?25–26 Find the approximations , , , and for , 10 and 20. Then compute the corresponding

Let , where is the function whose graph is shown. (a) Use the graph to find and . (b) Are these underestimates or overestimates of ? (c) Use the graph to find . How does it compare with ? (d) For any value of , list the numbers , , , , and in increasing order.

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate , where is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each case. (a) Which rule produced which estimate? (b) Between which two approximations does the true value of lie?

Estimate using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with n = 4. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?

Draw the graph of in the viewing rectangle [1, 0] by [0, 0.5] and let. (a) Use the graph to decide whether , , and underestimate or overestimate . (b) For any value of , list the numbers , , , , and in increasing order. (c) Compute , , , and . From the graph, which do you think gives the best estimate of I?

5–6 Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. 5.

5–6 Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. 6.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 7.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 8.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 9.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 10.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 11.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 12.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 13.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 14.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 15.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 16.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 17.

7–18 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of . (Round your answers to six decimal places.) 18.

(a) Find the approximations and for the integral . (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose so that the approximations and to the integral in part (a) are accurate to within 0.0001?

(a) Find the approximations and for . (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose so that the approximations and to the integral in part (a) are accurate to within 0.0001?

(a) Find the approximations , , and for and the corresponding errors , , and . (b) Compare the actual errors in part (a) with the error estimates given by 3 and 4. (c) How large do we have to choose so that the approximations , , and to the integral in part (a) are accurate to within 0.00001?

How large should be to guarantee that the Simpson’s Rule approximation to is accurate to within 0.00001?

The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound for by hand. But computer algebra systems have no problem computing and graphing it, so we can easily find a value for K from a machine graph. This exercise deals with approximations to the integral , where . (a) Use a graph to get a good upper bound for . (b) Use to approximate I. (c) Use part (a) to estimate the error in part (b). (d) Use the built-in numerical integration capability of your CAS to approximate . (e) How does the actual error compare with the error estimate in part (c)? (f ) Use a graph to get a good upper bound for . (g) Use to approximate . (h) Use part (f ) to estimate the error in part (g). (i) How does the actual error compare with the error estimate in part (h)? ( j) How large should be to guarantee that the size of the error in using is less than ?

Repeat Exercise 23 for the integral .

25–26 Find the approximations , , , and for , 10 and 20. Then compute the corresponding errors , , , and . (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when is doubled? 25.

25–26 Find the approximations , , , and for , 10 and 20. Then compute the corresponding errors , , , and . (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when is doubled? 26.

27–28 Find the approximations , , and for and 12. Then compute the corresponding errors , , and . (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when is doubled? 27.

27–28 Find the approximations , , and for and 12. Then compute the corresponding errors , , and . (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when is doubled? 28.

Estimate the area under the graph in the figure by using (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule, each with .

The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool.

(a) Use the Midpoint Rule and the given data to estimate the value of the integral . (b) If it is known that for all , estimate the error involved in the approximation in part (a).

(a) A table of values of a function is given. Use Simpson’s rule to estimate . (b) If for , estimate the error involved in the approximation in part (a).

A graph of the temperature in New York City on September 19, 2009 is shown. Use Simpson’s Rule with n = 12 to estimate the average temperature on that day.

A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see the table). Use Simpson’s Rule to estimate the distance the runner covered during those 5 seconds.

The graph of the acceleration of a car measured in is shown. Use Simpson’s Rule to estimate the increase in the velocity of the car during the 6-second time interval.

Water leaked from a tank at a rate of liters per hour, where the graph of is as shown. Use Simpson’s Rule to estimate the total amount of water that leaked out during the first 6 hours.

The table (supplied by San Diego Gas and Electric) gives the power consumption in megawatts in San Diego County from midnight to 6:00 AM on a day in December. Use Simpson’s Rule to estimate the energy used during that time period. (Use the fact that power is the derivative of energy.)

Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 AM. is the data throughput, measured in megabits per second. Use Simpson’s Rule to estimate the total amount of data transmitted during that time period.

Use Simpson’s Rule with to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the -axis and (b) the -axis.

The table shows values of a force function , where is measured in meters and in newtons. Use Simpson’s Rule to estimate the work done by the force in moving an object a distance of 18 m.

The region bounded by the curves , , , and is rotated about the -axis. Use Simpson’s Rule with to estimate the volume of the resulting solid.

The figure shows a pendulum with length that makes a maximum angle with the vertical. Using Newton’s Second Law, it can be shown that the period (the time for one complete swing) is given by where and is the acceleration due to gravity. If m and =42°, use Simpson’s Rule with to find the period.

The intensity of light with wavelength traveling through a diffraction grating with slits at an angle is given by , where and is the distance between adjacent slits. A helium-neon laser with wavelength m is emitting a narrow band of light, given by , through a grating with 10,000 slits spaced m apart. Use the Midpoint Rule with to estimate the total light intensity emerging from the grating.

Use the Trapezoidal Rule with to approximate . Compare your result to the actual value. Can you explain the discrepancy?

Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with is more accurate than the Midpoint Rule.

Sketch the graph of a continuous function on [0, 2] for which the right endpoint approximation with is more accurate than Simpson’s Rule.

If is a positive function and for , show that

Show that if is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of .

Show that .

Show that .