Solved: Let be the rate at which the worlds oil is consumed, where is measured in years

(a) Write an expression for a Riemann sum of a function . 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.

(a) Write the denition of the denite integral of a continuous function from to . (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.

(a) State the Evaluation Theorem. (b) State the Net Change Theorem.

If is the rate at which water ows into a reservoir, what does represent?

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) Explain the meaning of the indenite integral . (b) What is the connection between the denite integral and the indenite integral ?

State both parts of the Fundamental Theorem of Calculus.

(a) State the Substitution Rule. In practice, how do you use it?(b) State the rule for integration by parts. In practice, how do you use it?

State the rules for approximating the denite integral with the Midpoint Rule, the Trapezoidal Rule, and Simpsons Rule. Which would you expect to give the best estimate? How do you approximate the error for each rule?

Dene the following improper integrals. (a) (b) (c) y  fx dxy

Dene the improper integral for each of the following cases. (a) has an innite discontinuity at . (b) has an innite discontinuity at . (c) has an innite discontinuity at , where .

State the Comparison Theorem for improper integrals.

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

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. All continuous functions have antiderivatives.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. All continuous functions have derivatives.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The Midpoint Rule is always more accurate than the Trapezoidal Rule.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is continuous, then .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is continuous on and is convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is a continuous, decreasing function on and , then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are both convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are both divergent, then is divergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and diverges, then also diverges.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is continuous on , then d dxyb a fxdx fx a, b

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the indenite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).

Evaluate the indenite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take ).

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

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

Find the derivative of the function.

Find the derivative of the function.

Find the derivative of the function.

Find the derivative of the function.

Use the Table of Integrals on the Reference Pages to evaluate the integral.

Use the Table of Integrals on the Reference Pages to evaluate the integral.

Use the Table of Integrals on the Reference Pages to evaluate the integral.

Use the Table of Integrals on the Reference Pages to evaluate the integral.

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule with to approximate the given integral. Round your answers to six decimal places. Can you say whether your answers are underestimates or overestimates?

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule with to approximate the given integral. Round your answers to six decimal places. Can you say whether your answers are underestimates or overestimates?

Estimate the errors involved in Exercise 47, parts (a) and (b). How large should be in each case to guarantee an error of less than 0.00001?

Use Simpsons Rule with to estimate the area under the curve from to .

(a) If , use a graph to nd an upper bound for . (b) Use Simpsons Rule with to approximate and use part (a) to estimate the error(c) How large should be to guarantee that the size of the error in using is less than ?

(a) How would you evaluate by hand? (Dont actually carry out the integration.) (b) How would you evaluate using tables? (Dont actually do it.) (c) Use a CAS to evaluate . (d) Graph the integrand and the indenite integral on the same screen.

Use Property 8 of integrals to estimate the value of

Use the properties of integrals to verify that 0 y1 0 x4 cos x dx  0.2

Evaluate the integral or show that it is divergent.

Evaluate the integral or show that it is divergent.

Evaluate the integral or show that it is divergent.

Evaluate the integral or show that it is divergent.

Evaluate the integral or show that it is divergent.

Evaluate the integral or show that it is divergent.

Use the Comparison Theorem to determine whether the integral y 1 x3 x5  2 dx is convergent or divergent.

For what values of is convergent? Use the Table of Integrals to evaluate the integral for those values of .

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 .

The speedometer reading ( ) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpsons Rule to estimate the distance traveled by the car.

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

Suppose that the temperature in a long, thin rod placed along the -axis is initially if and 0 if . It can be shown that if the heat diffusivity of the rod is , then the temperature of the rod at the point at time is Tx, t  C as4  kt ya 0 exu24kt du To nd the temperature distribution that results from an initial hot spot concentrated at the origin, we need to compute lim al0 Tx, t Use lHospitals Rule to nd this limit.

The Fresnel function was introduced in Section 5.4. Fresnel also used the function Cx yx 0 cos(1 2  t2)dt in his theory of the diffraction of light waves. (a) On what intervals is increasing? (b) On what intervals is concave upward? (c) Use a graph to solve the following equation correct to two decimal places: yx 0 cos(1 2  t2)dt  0.7 (d) Plot the graphs of and on the same screen. How are these graphs related?

If is a continuous function such that yx 0 ftdt  xe2x yx 0 etftdt for all , nd an explicit formula for .

Find a function and a value of the constant such that 2 yx a ft dt  2 sin x  1

If is continuous on , show that 2 yb a fxfxdx  fb2  fa2

If is a positive integer, prove that y1 0 ln xn dx  1nn!

If is continuous on and , show that y 0 fx dx  f0

The gure shows two regions in the rst quadrant: is the area under the curve from to , and is the area of the triangle with vertices , , and . Find .