Bagel output The manager of a bagel bakery collects the

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume f and f_ are continuous functions for all real numbers. a. If A1x2 = 1 x a f 1t2 dt and f 1t2 = 2t - 3, then A is a quadratic function. b. Given an area function A1x2 = 1 x a f 1t2 dt and an antiderivative F of f , it follows that A_1x2 = F 1x2. c. 1 b a f _1x2 dx = f 1b2 - f 1a2. d. If f is continuous on 3a, b4 and 1 b a _ f 1x2 _ dx = 0, then f 1x2 = 0 on 3a, b4. e. If the average value of f on 3a, b4 is zero, then f 1x2 = 0 on 3a, b4. f. 1 b a 12f 1x2 - 3g1x22 dx = 21 b a f 1x2 dx + 31 a b g1x2 dx. g. 1f _1g1x22g_1x2 dx = f 1g1x22 + C.

Velocity to displacement An object travels on the x-axis with a velocity given by v1t2 = 2t + 5, for 0 t 4. a. How far does the object travel, for 0 t 4? b. What is the average value of v on the interval 30, 44? c. True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant), for 0 t 4.

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of f is given in the figure. a. L 4 0 f 1x2 dx b. L 4 6 f 1x2 dx c. L 7 5 f 1x2 dx d. L 7 0 f 1x2 dx

Displacement by geometry Use geometry to find the displacement of an object moving along a line for the time intervals (i) 0 t 5, (ii) 3 t 7, and (iii) 0 t 8, where the graph of its velocity v = g1t2 is given in the figure.

Area by geometry Use geometry to evaluate 1 4 0 28x - x2 dx. (Hint: Complete the square.)

Bagel output The manager of a bagel bakery collects the following production rate data (in bagels per minute) at seven different times during the morning. Estimate the total number of bagels produced between 6:00 and 7:30 a.m., using a left and right Riemann sum. Time of day (a.m.) Production rate (bagels/min)

Integration by Riemann sums Consider the integral 1 4 1 13x - 22 dx. a. Evaluate the right Riemann sum for the integral with n = 3. b. Use summation notation to express the right Riemann sum in terms of a positive integer n. c. Evaluate the definite integral by taking the limit as nS _ of the Riemann sum of part (b). d. Confirm the result of part (c) by graphing y = 3x - 2 and using geometry to evaluate the integral. Then evaluate 1 4 1 13x - 22 dx with the Fundamental Theorem of Calculus.

811. Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. L 1 0 14x - 22 dx

811. Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. L 2 0 1x2 - 42 dx

811. Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. L 2 1 13x2 + x2 dx

811. Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. L 4 0 1x3 - x2 dx

Evaluating Riemann sums Consider the function f 1x2 = 3x + 4 on the interval 33, 74. Show that the midpoint Riemann sum with n = 4 gives the exact area of the region bounded by the graph.

Sum to integral Evaluate the following limit by identifying the integral that it represents: lim nS_ a n k = 1 aa 4k n b 5 + 1b a 4 n b.

Area function by geometry Use geometry to find the area A1x2 that is bounded by the graph of f 1t2 = 2t - 4 and the t-axis between the point 12, 02 and the variable point 1x, 02, where x 2. Verify that A_1x2 = f 1x2.

1530. Evaluating integrals Evaluate the following integrals. L 2 -2 13x4 - 2x + 12 dx

1530. Evaluating integrals Evaluate the following integrals. L cos 3x dx

1530. Evaluating integrals Evaluate the following integrals. L 2 0 1x + 123 dx

1530. Evaluating integrals Evaluate the following integrals. L 1 0 14x21 - 2x16 + 12 dx

1530. Evaluating integrals Evaluate the following integrals. L 19x8 - 7x62 dx

1530. Evaluating integrals Evaluate the following integrals. L 2 -2 e4x + 8 dx

1530. Evaluating integrals Evaluate the following integrals. L 1 0 1x11x + 12 dx

1530. Evaluating integrals Evaluate the following integrals. L y2 y3 + 27 dy

1530. Evaluating integrals Evaluate the following integrals. L 1 0 dx 24 - x2

1530. Evaluating integrals Evaluate the following integrals. L y213y3 + 124 dy

1530. Evaluating integrals Evaluate the following integrals. L 3 0 x 225 - x2 dx

1530. Evaluating integrals Evaluate the following integrals. L x sin x2 cos8 x2 dx

1530. Evaluating integrals Evaluate the following integrals. L p 0 sin 2 5u du

1530. Evaluating integrals Evaluate the following integrals. L p 0 11 - cos2 3u2 du

1530. Evaluating integrals Evaluate the following integrals. L 3 2 x2 + 2x - 2 x3 + 3x2 - 6x dx

1530. Evaluating integrals Evaluate the following integrals. L ln 2 0 ex 1 + e2x dx

3134. Area of regions Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = 16 - x2 on 3-4, 44

3134. Area of regions Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = x3 - x on 3-1, 04

3134. Area of regions Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = 2 sin1x>42 on 30, 2p4

3134. Area of regions Compute the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = 1>1x2 + 12 on 3-1, 234

3536. Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = x4 - x2 on 3-1, 14

3536. Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of f and the x-axis on the given interval. You may find it useful to sketch the region. f 1x2 = x2 - x on 30, 34

Symmetry properties Suppose that 1 4 0 f 1x2 dx = 10 and 1 4 0 g1x2 dx = 20. Furthermore, suppose that f is an even function and g is an odd function. Evaluate the following integrals. a. L 4 -4 f 1x2 dx b. L 4 -4 3g1x2 dx c. L 4 -4 14f 1x2 - 3g1x22 dx d. L 1 0 8x f 14x22 dx e. L 2 -2 3x f 1x2 dx

Properties of integrals The figure shows the areas of regions bounded by the graph of f and the x-axis. Evaluate the following integrals. a. L c a f 1x2 dx b. L d b f 1x2 dx c. L b c 2 f 1x2 dx d. L d a 4 f 1x2 dx e. L b a 3 f 1x2 dx f. L d b 2 f 1x2 dx y a b c d x Area _ 20 y _ f (x) Area _ 12 Area _ 15

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. L 4 1 3 f 1x2 dx

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. - L 1 4 2 f 1x2 dx

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. L 4 1 13 f 1x2 - 2g1x22 dx

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. L 4 1 f 1x2g1x2 dx

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. L 3 1 f 1x2 g1x2 dx

3944. Properties of integrals Suppose that 1 4 1 f 1x2 dx = 6, 1 4 1 g1x2 dx = 4, and 1 4 3 f 1x2 dx = 2. Evaluate the following integrals or state that there is not enough information. L 1 4 1 f 1x2 - g1x22 dx

Displacement from velocity A particle moves along a line with a velocity given by v1t2 = 5 sin pt starting with an initial position s102 = 0. Find the displacement of the particle between t = 0 and t = 2, which is given by s1t2 = 1 2 0 v1t2 dt. Find the distance traveled by the particle during this interval, which is 1 2 0 _ v1t2 _ dt.

Average height A baseball is launched into the outfield on a parabolic trajectory given by y = 0.01x1200 - x2. Find the average height of the baseball over the horizontal extent of its flight.

Average values Integration is not needed. a. Find the average value of f shown in the figure on the interval 31, 64 and then find the point(s) c in 11, 62 guaranteed to exist by the Mean Value Theorem for Integrals. b. Find the average value of f shown in the figure on the interval 32, 64 and then find the point(s) c in 12, 62 guaranteed to exist by the Mean Value Theorem for Integrals.

An unknown function The function f satisfies the equation 3x4 - 48 = 1 x 2 f 1t2 dt. Find f and check your answer by substitution.

An unknown function Assume f _ is continuous on 32, 44, 1 2 1 f _12x2 dx = 10, and f 122 = 4. Evaluate f 142.

Function defined by an integral Let H1x2 = 1 x 0 24 - t2 dt, for -2 x 2. a. Evaluate H102. b. Evaluate H_112. c. Evaluate H_122. d. Use geometry to evaluate H122. e. Find the value of s such that H1x2 = sH1-x2.

Function defined by an integral Make a graph of the function f 1x2 = , for x 1. Be sure to include all of the evidence you used to arrive at the graph.

Identifying functions Match the graphs A, B, and C in the figure with the functions f 1x2, f _1x2, and 1

Ascent rate of a scuba diver Divers who ascend too quickly in the water risk decompression illness. A common recommendation for a maximum rate of ascent is 30 feet/minute with a 5-minute safety stop 15 feet below the surface of the water. Suppose that a diver ascends to the surface in 8 minutes according to the velocity function v1t2 = 30 if 0 t 2 0 if 2 6 t 7 15 if 7 6 t 8. a. Graph the velocity function v. b. Compute the area under the velocity curve. c. Interpret the physical meaning of the area under the velocity curve.

Area functions Consider the graph of the continuous function f in the figure and let F1x2 = L x 0 f 1t2 dt and G1x2 = L x 1 f 1t2 dt. Assume the graph consists of a line segment from 10, -22 to 12, 22 and two quarter circles of radius 2. 2 1 _1 _2 y _ f (t) 1 2 3 4 t y _2 _1 a. Evaluate F122, F1-22, and F142. b. Evaluate G1-22, G102, and G142. c. Explain why there is a constant C such that F1x2 = G1x2 + C, for -2 x 4. Fill in the blank with a number: F1x2 = G1x2 + _____, for -2 x 4.

5556. Area functions and the Fundamental Theorem Consider the function Let F1x2 = L x -1 f 1t2 dt and G1x2 = L x -2 f 1t2 dt.a. Evaluate F1-22 and F122. b. Use the Fundamental Theorem to find an expression for F_1x2, for -2 x 6 0. c. Use the Fundamental Theorem to find an expression for F_1x2, for 0 x 2. d. Evaluate F_1-12 and F_112. Interpret these values. e. Evaluate F_1-12 and F_112. f. Find a constant C such that F1x2 = G1x2 + C.

5556. Area functions and the Fundamental Theorem Consider the function Let F1x2 = L x -1 f 1t2 dt and G1x2 = L x -2 f 1t2 dt.a. Evaluate G1-12 and G112. b. Use the Fundamental Theorem to find an expression for G_1x2, for -2 x 6 0. c. Use the Fundamental Theorem to find an expression for G_1x2, for 0 x 2. d. Evaluate G_102 and G_112. Interpret these values. e. Find a constant C such that F1x2 = G1x2 + C.

5758. Limits with integrals Evaluate the following limits. lim xS2 1 x 2 et 2 dt x - 2

5758. Limits with integrals Evaluate the following limits. lim xS1 1 x2 1 et 3 dt x - 1

Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n: L 1 0 xn dx + L 1 0 1n x dx = 1.

Change of variables Use the change of variables u3 = x2 - 1 to evaluate the integral 1 3 1 x 23 x2 - 1 dx.

Inverse tangent integral Prove that for nonzero constants a and b, L dx a2x2 + b2 = 1 ab tan-1 a ax b b + C.

6267. Additional integrals Evaluate the following integrals. L sin 2x 1 + cos2 x dx 1Hint: sin 2x = 2 sin x cos x.2

6267. Additional integrals Evaluate the following integrals. L 1 x2 sin 1 x dx

6267. Additional integrals Evaluate the following integrals. L 1tan-1 x25 1 + x2 dx

6267. Additional integrals Evaluate the following integrals. L dx 1tan-1 x211 + x22

6267. Additional integrals Evaluate the following integrals. L sin-1 x 21 - x2 dx

6267. Additional integrals Evaluate the following integrals. L ex - e-x ex + e-x dx

Area with a parameter Let a 7 0 be a real number and consider the family of functions f 1x2 = sin ax on the interval 30, p>a4. a. Graph f , for a = 1, 2, 3. b. Let g1a2 be the area of the region bounded by the graph of f and the x-axis on the interval 30, p>a4. Graph g for 0 6 a 6 _. Is g an increasing function, a decreasing function, or neither?

Equivalent equations Explain why if a function u satisfies the equation u1x2 + 21 x 0 u1t2 dt = 10, then it also satisfies the equation u_1x2 + 2u1x2 = 0. Is it true that if u satisfies the second equation, then it satisfies the first equation?

Area function properties Consider the function f 1t2 = t2 - 5t + 4 and the area function A1x2 = 1 x 0 f 1t2 dt. a. Graph f on the interval 30, 64. b. Compute and graph A on the interval 30, 64. c. Show that the local extrema of A occur at the zeros of f . d. Give a geometrical and analytical explanation for the observation in part (c). e. Find the approximate zeros of A, other than 0, and call them x1 and x2, where x1 6 x2. f. Find b such that the area bounded by the graph of f and the t-axis on the interval 30, t14 equals the area bounded by the graph of f and the t-axis on the interval 3t1, b4. g. If f is an integrable function and A1x2 = 1 x a f 1t2 dt, is it always true that the local extrema of A occur at the zeros of f ? Explain.

Function defined by an integral Let f 1x2 = 1 x 0 1t - 12151t - 229 dt. a. Find the intervals on which f is increasing and the intervals on which f is decreasing. b. Find the intervals on which f is concave up and the intervals on which f is concave down. c. For what values of x does f have local minima? Local maxima? d. Where are the inflection points of f ?

Exponential inequalities Sketch a graph of f 1t2 = et on an arbitrary interval 3a, b4. Use the graph and compare areas of regions to prove that e1a + b2>2 6 eb - ea b - a 6 ea + eb 2 .