Solution: Use Part 1 of the Fundamental Theorem of Calculus

Evaluate the integral. y du 2 0 6x 2 4x 5 dx y 3

Evaluate the integral. u5 u3 u2 y du 2 0 6

Evaluate the integral. 0 x45 dx

Evaluate the integral. y x dx 1

Evaluate the integral. x2 x5 d

Evaluate the integral. y 2  y cos  d

Evaluate the integral. y 2v 53v 1 dv 2 2 3u 1 2 du y 2 

Evaluate the integral. y 4 0 y 2v 53v 1 dv 2 2

Evaluate the integral. 1 2 4y 3 2 y 3 dy y

Evaluate the integral. y 2 1 y 5y7 y 3 y dy 1

Evaluate the integral. y dx 1 0 x(s 3 x s 4 13. x ) dx

Evaluate the integral. y 9 1 3x 2 sx y dx 1

Evaluate the integral. y (3 xsx ) dx 4 0 sec2 t dt

Evaluate the integral. y 1 0 y (3 xsx ) dx 

Evaluate the integral. y csc  cot  d 2  csc2  d

Evaluate the integral. csc  cot  d

Evaluate the integral. 1 0 (s 4 x 5 s 5 x 4 ) dx

Evaluate the integral. 0 s2t dt

Evaluate the integral. 1 s 3 x sx dx

Evaluate the integral. y 3 4 y sec  tan  d

Evaluate the integral. 1  5 x dx

Evaluate the integral. y 1 0 1 x 2 3 y dx 4 1

Evaluate the integral. y d 4 0 1 cos2  cos2  d

Evaluate the integral. 3 0 sin  sin  tan2  sec2  y d 

Evaluate the integral. 1 (x 2  x  27. ) dx

Evaluate the integral. 2 0  sin x y  dx 2

What is wrong with the equation? y 3 1 1 x 2 dx x1 1  3 1 4 3 y 3

What is wrong with the equation? 0 sec2 x dx tan x]0  0 y

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y sin x, 0 x 

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y sec 0 x 3 2

Use a graph to estimate the -intercepts of the curve . Then use this information to estimate the area of the region that lies under the curve and above the -axis.

Repeat Exercise 33 for the curve .

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. 2 1 x 3 35. dx

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. y 52 4 y sin x dx

Verify by differentiation that the formula is correct. sx 2 1 dx sx 2 1 C y

Verify by differentiation that the formula is correct. x cos x dx x sin x cos x C y

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. xsx dx

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. cos x 2 sin x dx3

Find the general indefinite integral 1 t2 t dx 2 41. dt y xsx

Find the general indefinite integral x1 2x 4 y 1 t2 t dx 2 41. d

Find the general indefinite integral sin x 1 sin2 x dx

Find the general indefinite integral y sin 2x sin x y dx

The area of the region that lies to the right of the -axis and to the left of the parabola (the shaded region in the figure) is given by the integral . (Turn your head clockwise and think of the region as lying below the curve from to .) Find the area of the region.

The boundaries of the shaded region are the y-axis, the line , and the curve . Find the area of this region by writing x as a function of y and integrating with respect to y (as in Exercise 45).

If is the rate of growth of a child in pounds per year, what does represent?

The current in a wire is defined as the derivative of the charge: . What does represent?

If oil leaks from a tank at a rate of gallons per minute at time , what does represent?

A honeybee population starts with 100 bees and increases at a rate of bees per week. What does represent?

In Section 3.5 we defined the marginal revenue function as the derivative of the revenue function , where is the number of units sold. What does represent?

If is the slope of a trail at a distance of miles from the start of the trail, what does represent?

If is measured in meters and is measured in newtons, what are the units for ?

If the units for are feet and the units for are pounds per foot, what are the units for ? What units does have?

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. vt 3t 5 0 t 3 x8

The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. vt t 1 t 6 2 2t 8 v

The acceleration function (in ms ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time and (b) the distance traveled during the given time interval. at t 4 v0 5 0 t 10 t 2 Th

The acceleration function (in ms ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time and (b) the distance traveled during the given time interval. at 2t 3 v0 4 0 t 3 at

The velocity of a car was read from its speedometer at 10-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car.

Suppose that a volcano is erupting and readings of the rate at which solid materials are spewed into the atmosphere are given in the table. The time is measured in seconds and the units for are tonnes (metric tons) per second. (a) Give upper and lower estimates for the total quantity of erupted materials after 6 seconds. (b) Use the Midpoint Rule to estimate .

Water flows from the bottom of a storage tank at a rate of liters per minute, where . Find the amount of water that flows from the tank during the first 10 minutes.

Water flows into and out of a storage tank. A graph of the rate of change of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time is 25,000 L, use the Midpoint Rule to estimate the amount of water four days later.

Let , where is the function whose graph is shown. (a) Evaluate , , , , and . (b) On what interval is increasing? (c) Where does have a maximum value? (d) Sketch a rough graph of

Let , where is the function whose graph is shown. (a) Evaluate for and 6. (b) Estimate . (c) Where does have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of .

Sketch the area represented by . Then find in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. tx y (1 st ) dt x 0 1 t 2 dt tx tx

Sketch the area represented by . Then find in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating. tx y x 0 tx y (1 st ) dt x 0 1

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tx y dt x 0 s1 2t dt tx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. x y x 1 2 t 4 5 tx y dt x 0 s1

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. t y y y 2 t 2 7. sin t dt tx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. Fx y 10 x tan  d t Hint:y 10 x tan  d y x 10 tan  d

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. hx y dr 1x 2 sin4 9. t dt Let

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. hx y x 2 0 s1 r 3 hx y dr 1x 2

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y y t dt sx 3 cos t t dt

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y y 0 1x 2 sin3 y y t dt sx

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tx y 3x 2x u2 1 u2 1 13. du y yHint:y 3x 2x f u du y 0 2x f u du y 3x 0 f u du tx y

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y y cos x sin x 1 v 2 10 dv  H

Find the average value of the function on the given interval. f x x , 0, 2 2 , 1, 1 y

Find the average value of the function on the given interval. f x x x 2 f x x , 0, 2 2 , 1, 1

Find the average value of the function on the given interval. tx cos x, 0, 2 f x

Find the average value of the function on the given interval. f  sec  tan , 0, 4 tx

(a) Find the average value of on the given interval. (b) Find such that . (c) Sketch the graph of and a rectangle whose area is the same as the area under the graph of . f x x 3 2, 5 2 19. f f

(a) Find the average value of on the given interval. (b) Find such that . (c) Sketch the graph of and a rectangle whose area is the same as the area under the graph of . fx sx 0, 4 f

The table gives values of a continuous function. Use the Midpoint Rule to estimate the average value of on .

The velocity graph of an accelerating car is shown. (a) Estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity?

If , where , find .

Find the interval on which the curve is concave upward.

Let , where is the function whose graph is shown. (a) At what values of do the local maximum and minimum values of occur? (b) Where does attain its absolute maximum value? (c) On what intervals is concave downward? (d) Sketch the graph of .

Let , where is the function whose graph is shown. (a) At what values of do the local maximum and minimum values of occur? (b) Where does attain its absolute maximum value? (c) On what intervals is concave downward? (d) Sketch the graph of .

If , is continuous, and , what is the value of ?

Suppose h is a function such that , , , , , , and is continuous everywhere. Evaluate .

The Fresnel function was defined in Example 4 and graphed in Figures 7 and 8. (a) At what values of does this function have local maximum values? (b) On what intervals is the function concave upward? (c) Use a graph to solve the following equation correct to two decimal places: y x 0 sint 2 2 dt 0.2 CA

The sine integral function is important in electrical engineering. [The integrand is not defined when , but we know that its limit is 1 when . So we define and this makes a continuous function everywhere.] (a) Draw the graph of . (b) At what values of does this function have local maximum values? (c) Find the coordinates of the first inflection point to the right of the origin. (d) Does this function have horizontal asymptotes? (e) Solve the following equation correct to one decimal place: y x 0 sin t t dt 1 x

Find a function and a number such that for all

A high-tech company purchases a new computing system whose initial value is . The system will depreciate at the rate and will accumulate maintenance costs at the rate , where is the time measured in months. The company wants to determine the optimal time to replace the system. (a) Let Show that the critical numbers of occur at the numbers where . (b) Suppose that and Determine the length of time for the total depreciation to equal the initial value . (c) Determine the absolute minimum of on . (d) Sketch the graphs of and in the same coordinate system, and verify the result in part (a) in this case.

A manufacturing company owns a major piece of equip- C Ct ment that depreciates at the (continuous) rate , where is the time measured in months since its last overhaul. Because a fixed cost is incurred each time the machine is overhauled, the company wants to determine the optimal time (in months) between overhauls. (a) Explain why represents the loss in value of the machine over the period of time since the last overhaul. t ) Let be given by What does represent and why would the company want to minimize ? (c) Show that has a minimum value at the numbers where . CT fT C t

Evaluate the integral by making the given substitution. y 4 1 2x 3 dx, u 1 2x y si

Evaluate the integral by making the given substitution. y cos4  sin  d, u cos 

Evaluate the indefinite integral. 2xx dx 2 3 4 dx y

Evaluate the indefinite integral. x 2 x 3 5 9 y 2xx dx 2 3

Evaluate the indefinite integral. y 3x 2 dx 20 dx y

Evaluate the indefinite integral. 2 x 6 y 3x 2 dx 20 dx

Evaluate the indefinite integral. y dx 3 2y 1 5 dy y 2

Evaluate the indefinite integral. sx sin1 x 32 y dx 3

Evaluate the indefinite integral. cos st st 13. dt

Evaluate the indefinite integral. x x 2 1 2 y dx cos

Evaluate the indefinite integral. a bx 2 s3ax bx 3 dx y

Evaluate the indefinite integral. 1 5t 4 2.7 y dt a

Evaluate the indefinite integral. in t dt

Evaluate the indefinite integral. y 3 s2y y sin t dt 4 1 dy

Evaluate the indefinite integral. s 3 1 z 3 dz

Evaluate the indefinite integral. sec 2 tan 2 d

Evaluate the indefinite integral. cos  sin  d 6  d

Evaluate the indefinite integral. 1 tan  5 sec2 y cos  sin  d 6

Evaluate the indefinite integral. scot x csc dx 2 23. x dx

Evaluate the indefinite integral. s 3 x 3 1 x 5 y scot x csc dx 2

Evaluate the indefinite integral. x c 0, a 1 a sb cx a1 25. dx 16 E

Evaluate the indefinite integral. y cosx x 2 dx y

Evaluate the indefinite integral. y sec cos t dt 3 x tan x dx y

Evaluate the indefinite integral. sin t sec2 y sec cos t dt 3

Evaluate the indefinite integral. s 4 x 2 dx

Evaluate the indefinite integral. s1 x y dx

Evaluate the definite integral. 2 0 x 1 25 dx y

Evaluate the definite integral. \7 0 y s4 3x dx 2

Evaluate the definite integral. 1 0 x 2 1 2x 3 5 dx y

Evaluate the definite integral. 0 x cosx 2 y dx 1

Evaluate the definite integral. sec2 t4 dt y

Evaluate the definite integral. \12 16 y csc t cot t dt

Evaluate the definite integral. 3 0 sin  cos2  37. d

Evaluate the definite integral. y 2 0 y cos x sinsin x dx

Evaluate the definite integral. y dx 2 1 xsx 1 dx

Evaluate the definite integral. y 2 2 x 2 sin x 1 x 6 y dx 2

Evaluate the definite integral. 13 0 dx s 3 1 2x2 y

Evaluate the definite integral. 4 0 x s1 2x y dx 1

Evaluate the definite integral. y 44. dx 6 6 tan3  d

Evaluate the definite integral. a 0 xsa2 x 2 y 44. dx

Find the average value of the function on the given interval. ft ts1 t 2 , 0, 5 y

Find the average value of the function on the given interval. tx x 2 s1 x 3 , 0, 2 ft

Find the average value of the function on the given interval. hx cos4 47. x sin x, 0,  tx

Find the average value of the function on the given interval. hr 31 r 2 , 1, 6 hx c

Evaluate by writing it as a sum of 53. f fx dx 10 two integrals and interpreting one of those integrals in terms of an area. 50

Evaluate by making a substitution and interpreting the resulting integral in terms of an area.

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 Ls. This explains, in part, why the function has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time .

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.

If is continuous and , find .

If is continuous and , find .

If is continuous on , prove that For the case where and , draw a diagram to interpret this equation geometrically as an equality of areas.

If is continuous on , prove that For the case where , draw a diagram to interpret this equation geometrically as an equality of areas.

If and are positive numbers, show that y 1 0 x a 1 x b dx y 1 0 x b 1 x a dx 57. a

(a) Write an expression for a Riemann sum of a function . x f x dx Explain the meaning of the notation that you use. (b) If , what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. 2

(a) Write the definition of the definite integral of from to . How does the definition simplify if you know that is continuous and you use equal subintervals? (b) What is the geometric interpretation of if ? (c) What is the geometric interpretation of if takes on both positive and negative values? Illustrate with a diagram.

State the Midpoint Rule.

(a) State the Evaluation Theorem. (b) State the Net Change Theorem. (c) If is the rate at which water flows into a reservoir, what does represent?

(a) Explain the meaning of the indefinite integral . (b) What is the connection between the definite integral and the indefinite integral ?

State both parts of the Fundamental Theorem of Calculus.

Suppose a particle moves back and forth along a straight line with velocity , measured in feet per second, and acceleration . (a) What is the meaning of ? (b) What is the meaning of ? (c) What is the meaning of ?

(a) What is the average value of a function on an interval ? (b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation?

Explain exactly what is meant by the statement that differentiation and integration are inverse processes.

State the Substitution Rule. In practice, how do you use it?

y 1 2 1 x 4 dx 3 8 y

0 x x 3 d12. represents the area under the curve y x x 3 x from 0 to 2.

All continuous functions have antiderivatives.

Evaluate the integral, if it exists. y 2 0 y 2 s1 y 3 y dy

Evaluate the integral, if it exists. 0 v2 cosv3 dv y

Evaluate the integral, if it exists. 0 y sin3t dt 1

Evaluate the integral, if it exists. sx 2 4x dx y

Evaluate the integral, if it exists. csc2 y 3t dt

Evaluate the integral, if it exists. sin t cos t dt

Evaluate the integral, if it exists. sin x coscos x dx y

Evaluate the integral, if it exists. 8 0 sec 2 tan 2 d

Evaluate the integral, if it exists. 4 0  sx 1  y dx

Evaluate the integral, if it exists. 3 0  x 2 4  dx

Evaluate the integral, if it exists. 1 sin x 1 x 2 y dx 3

Use a graph to give a rough estimate of the area of the region that lies under the curve . Then find the exact area.

Graph the function and use the graph to guess the value of the integral . Then evaluate the integral to confirm your guess.

Find the derivative of the function.Fx y x 1 s1 t 4 dt x2

Find the derivative of the function.tx y cos x 1 s 3 1 t 2 dt Fx

Find the derivative of the function.y y x sx cos   d

Find the derivative of the function.y y 3x1 2x sint 4 dt y

Use Property 8 of integrals to estimate the value of sx 2 3 dx

Use the properties of integrals to verify that 0 y 1 0 x 4 cos x dx 0.2

Use the Midpoint Rule with to approximate .

A particle moves along a line with velocity function , where is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval .

Let be the rate at which the worlds oil is consumed, where is measured in years starting at on January 1, 2000, and is measured in barrels per year. What does represent?

A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds.

A population of honeybees increased at a rate of bees per week, where the graph of is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.

Find the average value of the function on the interval .

If is a continuous function, what is the limit as of the average value of on the interval ?

. Let Evaluate by interpreting the integral as a difference of areas.

If is continuous on , prove that 1 0 f x dx y 1 0 f 1 x dx f 0,

If is a continuous function such that y x 0 f t dt x sin x y x 0 f t 1 t 2 dt f y for all , find an explicit formula for .

If is continuous on , show that 2 y b a f xf x dx f b 2 fa 2 f a, b

Evaluate lim n l 1 n  1 n  9  2 n  9  3 n  9  n n  9  2

(a) By reading values from the given graph of , use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of from to . In each case sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case.

(a) Use six rectangles to find estimates of each type for the area under the given graph of from to . (i) (sample points are left endpoints) (ii) (sample points are right endpoints) (iii) (sample points are midpoints) (b) Is an underestimate or overestimate of the true area? (c) Is an underestimate or overestimate of the true area? (d) Which of the numbers , , or gives the best estimate? Explain.

(a) Estimate the area under the graph of from to using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

(a) Estimate the area under the graph of from to using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

(a) Estimate the area under the graph of from to using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)(c), which appears to be the best estimate?

6. (a) Graph the function . (b) Estimate the area under the graph of using four approximating rectangles and taking the sample points to be (i) right endpoints (ii) midpoints In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles.

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.

Speedometer readings for a motorcycle at 12-second intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.

Oil leaked from a tank at a rate of liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.

When we estimate distances from velocity data, it is sometimes necessary to use times that are not equally spaced. We can still estimate distances using the time periods . For example, on May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the Earths surface of the space shuttle Endeavour, 62 seconds after liftoff.

The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.

The velocity graph of a car accelerating from rest to a speed of over a period of 30 seconds is shown. 1. Estimate the distance traveled during this period.

Use Definition 2 to find an expression for the area under the graph of as a limit. Do not evaluate the limit. fx s 1 x 16 4 x f

Use Definition 2 to find an expression for the area under the graph of as a limit. Do not evaluate the limit. f x 1 x 2 x 5 4 fx

Determine a region whose area is equal to Do not evaluate the limit.

. (a) Use Definition 2 to find an expression for the area under the curve from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first integers is proved in Appendix C. Use it to evaluate the limit in part (a). 13 23 33 n3  nn 1

(a) Express the area under the curve from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).

(a) Express the area under the curve from 2 to 7 as a limit. (b) Use a computer algebra system to evaluate the sum in part (a). (c) Use a computer algebra system to find the exact area by evaluating the limit of the expression in part (b).

Find the exact area under the cosine curve from to , where . (Use a computer algebra system both to evaluate the sum and to compute the limit.) In particular, what is the area if ? Compare your answer with the estimate obtained in Example 3(b).

(a) Let be the area of a polygon with equal sides inscribed in a circle with radius . By dividing the polygon into congruent triangles with central angle , show that . (b) Show that . [Hint: Use Equation 1.4.5 on page 42.]

Evaluate the Riemann sum for , with four equal subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

If , evaluate the Riemann sum for a regular partition with , taking the sample points to be left endpoints. What does the Riemann sum represent? Illustrate with a diagram.

If , find the Riemann sum for a regular partition with correct to six decimal places, n taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.

(a) Find the Riemann sum for , , with a regular partition and six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points.

Find the Riemann sum for , , if the partition points are and the sample points are .

Find the Riemann sum for , , if the partition points are and the sample points are left endpoints. What is ?

The graph of a function is given. Estimate using four subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints.

The graph of is shown. Estimate with six subintervals using (a) right endpoints, (b) left endpoints, and (c) midpoints.

A table of values of an increasing function is shown. Use the table to find lower and upper estimates for .

The table gives the values of a function obtained from an experiment. Use them to estimate using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be a decreasing function, can you say whether your estimates are less than or greater than the exact value of the integral?

Use the Midpoint Rule with the given value of to approximate the integral. Round the answer to four decimal places 2 sx 11. 3 1 dx, n 4 n

Use the Midpoint Rule with the given value of to approximate the integral. Round the answer to four decimal places 0 y secx3 dx, n 6 10 2 s

Use the Midpoint Rule with the given value of to approximate the integral. Round the answer to four decimal places y dx, n 4 1 0 sinx 2 dx, n 5 y

Use the Midpoint Rule with the given value of to approximate the integral. Round the answer to four decimal places y 5 1 x 1 x 1 y dx, n 4 1

Express the limit as a definite integral on the given interval lim nl  n i1 xi sin xi x, 0, 

Express the limit as a definite integral on the given interval lim nl  n i1 xi 1 xi x, 1, 5 l

Express the limit as a definite integral on the given interval lim 1, 8] max xil0  n i1 s2x*i x*i 17. 2 xi li

Express the limit as a definite integral on the given interval lim 0, 2 max xil0  n i1 4 3xi * 2 6xi * 5 xi lim 

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 5 1 1 3x dx lim

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 1 x y 2 2x 5 dx 5 1

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 0 2 x 2 21. dx y

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. y 5 0 1 2x 3 y dx 2 0

Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. y 2 1 x 3 dx

(a) Find an approximation to the integral using a Riemann sum with right endpoints and . (b) Draw a diagram like Figure 2 to illustrate the approximation in part (a). (c) Use Theorem 4 to evaluate . (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 6.

Express the integral as a limit of Riemann sums. Do not evaluate the limit. y sin x dx 6 2 x 1 x5 dx

Express the integral as a limit of Riemann sums. Do not evaluate the limit. y 2 0 x 2 y sin x dx

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. y dx  0 sin 5x d

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. y 10 2 x 6 y dx

The graph of is shown. Evaluate each integral by interpreting it in terms of areas. (a) (b) (c) (d)

The graph of t consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) (b) (c)

Evaluate the integral by interpreting it in terms of areas. 0 (1 2 x 1 dx x

Evaluate the integral by interpreting it in terms of areas. 2 2 s4 x 2 y dx 3

Evaluate the integral by interpreting it in terms of areas. y 3 2x dx 0 3 (1 s9 x 2 ) dx y 2

Evaluate the integral by interpreting it in terms of areas. y 3 1 y 3 2x dx 0 3

Evaluate the integral by interpreting it in terms of areas. 1  x  dx

Evaluate the integral by interpreting it in terms of areas. y 10 0  y x 5  dx 2

Given that , what is ?

Evaluate .

Write as a single integral in the form : 2 2 f x dx y 5 2 fx dx y 1 2 f x dx x b a 39

If and , find .

If and , find

Find if

In Example 2 in Section 4.1 we showed that . Use this fact and the properties of integrals to evaluate .

Suppose has absolute minimum value and absolute maximum value . Between what two values must lie? Which property of integrals allows you to make your conclusion?

Use the properties of integrals to verify that without evaluating the integral.

Use Property 8 to estimate the value of the integral. y dx 2 0 sx 3 1 dx

Use Property 8 to estimate the value of the integral. 1 1 x y dx

Use Property 8 to estimate the value of the integral. y tan x dx 2 0 x 3 3x 3 dx y

Use Property 8 to estimate the value of the integral. 4 y tan x dx 2

Use Property 8 to estimate the value of the integral. y 34 4 sin2 x dx

Express the following limit as a definite integral: lim n l  n i1 i 4 n5