Solved: Use the given graph of to find the Riemann sum

Use the given graph of to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents.

(a) Evaluate the Riemann sum for with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. (b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral (c) Use the Evaluation Theorem to check your answer to part (b). (d) Draw a diagram to explain the geometric meaning of the integral in part (b).

Evaluate by interpreting it in terms of areas.

Express as a definite integral on the interval and then evaluate the integral.

If and , find

(a) Write as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b) Use the Evaluation Theorem to check your answer to part (a).

The following figure shows the graphs of , and . Identify each graph, and explain your choices.

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 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 integral.

Evaluate the integral.

Evaluate the integral.

Evaluate the integral.

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

Evaluate the indefinite 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 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.

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 find an upper bound for . (b) Use Simpsons Rule with to approximate x 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 indefinite integral on the same screen.

Use Property 8 of integrals to estimate the value of

Use the properties of integrals to verify that

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 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 first 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 To find the temperature distribution that results from an initial hot spot concentrated at the origin, we need to compute Use lHospitals Rule to find this limit.

The Fresnel function was introduced in Section 5.4. Fresnel also used the function 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: (d) Plot the graphs of and on the same screen. How are these graphs related?

If is a continuous function such that for all , find an explicit formula for .

Find a function and a value of the constant such that

If is continuous on , show that

If is a positive integer, prove that

If is continuous on and , show tha

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