The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate y 2

Let I y 4 0 fsxd dx, where f is the function whose graph is shown. (a) Use the graph to find L2, R2, and M2. (b) Are these underestimates or overestimates of I? (c) Use the graph to find T2. How does it compare with I? (d) For any value of n, list the numbers Ln, Rn, Mn, Tn, and I in increasing order.

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate y 2 0 fsxd dx, where f 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 y 2 0 fsxd dx lie?

Estimate y 1 0 cossx 2 d dx 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 fsxd sin( 1 2 x 2 ) in the viewing rectangle f0, 1g by f0, 0.5g and let I y 1 0 fsxd dx. (a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate I. (b) For any value of n, list the numbers Ln, Rn, Mn, Tn, and I in increasing order. (c) Compute L5, R5, M5, and T5. From the graph, which do you think gives the best estimate of I?

56 Use (a) the Midpoint Rule and (b) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation.

56 Use (a) the Midpoint Rule and (b) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation.

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

718 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)

(a) Find the approximations T8 and M8 for the integral y 1 0 cos sx 2 d dx. (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001?

(a) Find the approximations T10 and M10 for y 2 1 e 1yx dx. (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001?

(a) Find the approximations T10, M10, and S10 for y  0 sin x dx and the corresponding errors ET, EM, and ES. (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 n so that the approximations Tn, Mn, and Sn to the integral in part (a) are accurate to within 0.00001?

How large should n be to guarantee that the Simpsons Rule approximation to y 1 0 ex 2 dx 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 K for | f s4d sxd | by hand. But computer algebra systems have no problem computing f s4d and graphing it, so we can easily find a value for K from a machine graph. This exercise deals with approximations to the integral I y 2 0 fsxd dx, where fsxd ecos x . (a) Use a graph to get a good upper bound for | f 0sxd |. (b) Use M10 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 I. (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 | f s4d sxd |. (g) Use S10 to approximate I. (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 n be to guarantee that the size of the error in using Sn is less than 0.0001?

Repeat Exercise 23 for the integral y 1 21 s4 2 x 3 dx.

Find the approximations Ln, Rn, Tn, and Mn for n 5, 10, and 20. Then compute the corresponding errors EL, ER, ET, and EM. (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 n is doubled?

Find the approximations Ln, Rn, Tn, and Mn for n 5, 10, and 20. Then compute the corresponding errors EL, ER, ET, and EM. (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 n is doubled?

2728 Find the approximations Tn, Mn, and Sn for n 6 and 12. Then compute the corresponding errors ET, EM, and ES. (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 n is doubled?

2728 Find the approximations Tn, Mn, and Sn for n 6 and 12. Then compute the corresponding errors ET, EM, and ES. (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 n is doubled?

Estimate the area under the graph in the figure by using (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsons Rule, each with n 6.

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

(a) Use the Midpoint Rule and the given data to estimate the value of the integral y 5 1 fsxd dx. x fsxd x fsxd 1.0 2.4 3.5 4.0 1.5 2.9 4.0 4.1 2.0 3.3 4.5 3.9 2.5 3.6 5.0 3.5 3.0 3.8 (b) If it is known that 22 < f 0sxd < 3 for all x, estimate the error involved in the approximation in part (a)

(a) A table of values of a function t is given. Use Simpsons Rule to estimate y 1.6 0 tsxd dx. x tsxd x tsxd 0.0 12.1 1.0 12.2 0.2 11.6 1.2 12.6 0.4 11.3 1.4 13.0 0.6 11.1 1.6 13.2 0.8 11.7 (b) If 25 < ts4d sxd < 2 for 0 < x < 1.6, estimate the error involved in the approximation in part (a).

A graph of the temperature in Boston on August 11, 2013, is shown. Use Simpsons 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 Simpsons

The graph of the acceleration astd of a car measured in ftys 2 is shown. Use Simpsons 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 rstd liters per hour, where the graph of r is as shown. Use Simpsons 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 P in megawatts in San Diego County from midnight to 6:00 am on a day in December. Use Simpsons 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 providers T1 data line from midnight to 8:00 am. D is the data throughput,

Use Simpsons Rule with n 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis.

Use Simpsons Rule with n 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis.

Use Simpsons Rule with n 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis.

Use Simpsons Rule with n 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis.

The intensity of light with wavelength  traveling through a diffraction grating with N slits at an angle  is given by Isd N 2 sin2 kyk 2 , where k sNd sin dy and d is the distance between adjacent slits. A helium-neon laser with wavelength  632.8 3 1029 m is emitting a narrow band of light, given by 21026 ,  , 1026 , through a grating with 10,000 slits spaced 1024 m apart. Use the Midpoint Rule with n 10 to estimate the total light intensity y1026 21026 Isd d emerging from the grating.

The intensity of light with wavelength  traveling through a diffraction grating with N slits at an angle  is given by Isd N 2 sin2 kyk 2 , where k sNd sin dy and d is the distance between adjacent slits. A helium-neon laser with wavelength  632.8 3 1029 m is emitting a narrow band of light, given by 21026 ,  , 1026 , through a grating with 10,000 slits spaced 1024 m apart. Use the Midpoint Rule with n 10 to estimate the total light intensity y1026 21026 Isd d emerging from the grating.

Sketch the graph of a continuous function on f0, 2g for which the Trapezoidal Rule with n 2 is more accurate than the Midpoint Rule.

Sketch the graph of a continuous function on f0, 2g for which the right endpoint approximation with n 2 is more accurate than Simpsons Rule

If f is a positive function and f 0sxd , 0 for a < x < b, show that Tn , y b a fsxd dx , Mn

Show that if f is a polynomial of degree 3 or lower, then Simpsons Rule gives the exact value of y b a fsxd dx.

Show that 1 2 sTn 1 Mn d T2n.

Show that 1 3Tn 1 2 3Mn S2n.