Stewart 8th Section 7.7: Approximate Integration

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Stewart 8th Section 7.7: Approximate Integration 0/57
1 Exercise - Approximation Methods · Level 3
Let \(I = \displaystyle\int_{0}^{4} f(x) d x\), where \(f\) is the function whose graph is shown. (a) Use the graph to find \(L_2\), \(R_2\), and \(M_2\). (b) Are these underestimates or overestimates of \(I\)? (c) Use the graph to find \(T_2\). How does it compare with \(I\)? (d) For any value of \(n\), list the numbers \(L_n\), \(R_n\), \(M_n\), \(T_n\), and \(I\) in increasing order.
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2 Exercise - Approximation Comparison · Level 3
The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate \(\displaystyle\int_{0}^{2} f(x) d x\), 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 \(\displaystyle\int_{0}^{2} f(x) d x\) lie?
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3 Exercise - Trapezoidal and Midpoint Rules · Level 2
Estimate \(\displaystyle\int_{0}^{1} \cos(x^2) d x\) 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?
4 Exercise - Approximation Analysis · Level 3
Draw the graph of \(f(x) = \sin\left(\dfrac{1}{2} x^2\right)\) in the viewing rectangle \([0, 1]\) by \([0, 0.5]\) and let \(I = \displaystyle\int_{0}^{1} f(x) d x\). (a) Use the graph to decide whether \(L_2\), \(R_2\), \(M_2\), and \(T_2\) underestimate or overestimate \(I\). (b) For any value of \(n\), list the numbers \(L_n\), \(R_n\), \(M_n\), \(T_n\), and \(I\) in increasing order. (c) Compute \(L_5\), \(R_5\), \(M_5\), and \(T_5\). From the graph, which do you think gives the best estimate of \(I\)?
5 Exercise - Midpoint and Simpson's Rules · Level 2
Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate \(\displaystyle\int_{0}^{2} \dfrac{x}{1+x^2} d x\) with \(n = 10\). (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation.
6 Exercise - Midpoint and Simpson's Rules · Level 2
Use (a) the Midpoint Rule and (b) Simpson's Rule to approximate \(\displaystyle\int_{0}^{\pi} x \cos x d x\) with \(n = 4\). (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approximation.
7 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{1}^{2} \sqrt{x^3 - 1} d x\) with \(n = 10\). (Round your answers to six decimal places.)
8 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{2} \dfrac{1}{1+x^6} d x\) with \(n = 8\). (Round your answers to six decimal places.)
9 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{2} \dfrac{e^x}{1+x^2} d x\) with \(n = 10\). (Round your answers to six decimal places.)
10 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sqrt[3]{1 + \cos x} d x\) with \(n = 4\). (Round your answers to six decimal places.)
11 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{4} x^3 \sin x d x\) with \(n = 8\). (Round your answers to six decimal places.)
12 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{1}^{3} e^{\dfrac{1}{x}} d x\) with \(n = 8\). (Round your answers to six decimal places.)
13 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{4} \sqrt{y} \cos y d y\) with \(n = 8\). (Round your answers to six decimal places.)
14 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{2}^{3} \dfrac{1}{\ln t} d t\) with \(n = 10\). (Round your answers to six decimal places.)
15 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{1} \dfrac{x^2}{1+x^4} d x\) with \(n = 10\). (Round your answers to six decimal places.)
16 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{1}^{3} \dfrac{\sin t}{t} d t\) with \(n = 4\). (Round your answers to six decimal places.)
17 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{4} \ln(1 + e^x) d x\) with \(n = 8\). (Round your answers to six decimal places.)
18 Exercise - All Three Rules · Level 2
Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate \(\displaystyle\int_{0}^{1} \sqrt{x + x^3} d x\) with \(n = 10\). (Round your answers to six decimal places.)
19 Exercise - Error Estimation · Level 3
(a) Find the approximations \(T_8\) and \(M_8\) for the integral \(\displaystyle\int_{0}^{1} \cos(x^2) d x\). (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose \(n\) so that the approximations \(T_n\) and \(M_n\) to the integral in part (a) are accurate to within \(0.0001\)?
20 Exercise - Error Estimation · Level 3
(a) Find the approximations \(T_{10}\) and \(M_{10}\) for \(\displaystyle\int_{1}^{2} e^{\dfrac{1}{x}} d x\). (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose \(n\) so that the approximations \(T_n\) and \(M_n\) to the integral in part (a) are accurate to within \(0.0001\)?
21 Exercise - Error Comparison · Level 3
(a) Find the approximations \(T_{10}\), \(M_{10}\), and \(S_{10}\) for \(\displaystyle\int_{0}^{\pi} \sin x d x\) and the corresponding errors \(E_T\), \(E_M\), and \(E_S\). (b) Compare the actual errors in part (a) with the error estimates given by the Trapezoidal/Midpoint and Simpson's Rule error bounds. (c) How large do we have to choose \(n\) so that the approximations \(T_n\), \(M_n\), and \(S_n\) to the integral in part (a) are accurate to within \(0.00001\)?
22 Exercise - Simpson's Rule Error · Level 3
How large should \(n\) be to guarantee that the Simpson's Rule approximation to \(\displaystyle\int_{0}^{1} e^{x^2} d x\) is accurate to within \(0.00001\)?
23 Exercise - CAS Error Bound Analysis · Level 4
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 \(|f^{(4)}(x)|\) by hand. But computer algebra systems have no problem computing \(f^{(4)}\) and graphing it, so we can easily find a value for \(K\) from a machine graph. This exercise deals with approximations to the integral \(I = \displaystyle\int_{0}^{2 \pi} f(x) d x\), where \(f(x) = e^{\cos x}\). (a) Use a graph to get a good upper bound for \(|f''(x)|\). (b) Use \(M_{10}\) 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 \(|f^{(4)}(x)|\). (g) Use \(S_{10}\) 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 \(S_n\) is less than \(0.0001\)?
24 Exercise - CAS Error Bound Analysis · Level 4
Repeat Exercise 23 for the integral \(\displaystyle\int_{-1}^1 \sqrt{4 - x^3} d x\).
25 Exercise - Approximations and Errors · Level 3
Find the approximations \(L_n\), \(R_n\), \(T_n\), and \(M_n\) for \(\displaystyle\int_{0}^{1} x e^x d x\) for \(n = 5\), \(10\), and \(20\). Then compute the corresponding errors \(E_L\), \(E_R\), \(E_T\), and \(E_M\). (Round your answers to six decimal places.) What observations can you make? In particular, what happens to the errors when \(n\) is doubled?
26 Exercise - Approximations and Errors · Level 3
Find the approximations \(L_n\), \(R_n\), \(T_n\), and \(M_n\) for \(\displaystyle\int_{1}^{2} \dfrac{1}{x^2} d x\) for \(n = 5\), \(10\), and \(20\). Then compute the corresponding errors \(E_L\), \(E_R\), \(E_T\), and \(E_M\). (Round your answers to six decimal places.) What observations can you make? In particular, what happens to the errors when \(n\) is doubled?
27 Exercise - Approximations and Errors · Level 3
Find the approximations \(T_n\), \(M_n\), and \(S_n\) for \(\displaystyle\int_{0}^{2} x^4 d x\) for \(n = 6\) and \(12\). Then compute the corresponding errors \(E_T\), \(E_M\), and \(E_S\). (Round your answers to six decimal places.) What observations can you make? In particular, what happens to the errors when \(n\) is doubled?
28 Exercise - Approximations and Errors · Level 3
Find the approximations \(T_n\), \(M_n\), and \(S_n\) for \(\displaystyle\int_{1}^{4} \dfrac{1}{\sqrt{x}} d x\) for \(n = 6\) and \(12\). Then compute the corresponding errors \(E_T\), \(E_M\), and \(E_S\). (Round your answers to six decimal places.) What observations can you make? In particular, what happens to the errors when \(n\) is doubled?
29 Exercise - Estimate Area from Graph · Level 2
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 \(n = 6\).
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30 Exercise - Simpson's Rule Area · Level 2
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.
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31 Exercise - Midpoint Rule with Data · Level 2
(a) Use the Midpoint Rule and the given data to estimate the value of the integral \(\displaystyle\int_{1}^{5} f(x) d x\). Data: \(f(1.0)=2.4\), \(f(1.5)=2.9\), \(f(2.0)=3.3\), \(f(2.5)=3.6\), \(f(3.0)=3.8\), \(f(3.5)=4.0\), \(f(4.0)=4.1\), \(f(4.5)=3.9\), \(f(5.0)=3.5\). (b) If it is known that \(-2 \leq f''(x) \leq 3\) for all \(x\), estimate the error involved in the approximation in part (a).
32 Exercise - Simpson's Rule with Data · Level 2
(a) A table of values of a function \(g\) is given. Use Simpson's Rule to estimate \(\displaystyle\int_{0}^{1.6} g(x) d x\). Data: \(g(0.0)=12.1\), \(g(0.2)=11.6\), \(g(0.4)=11.3\), \(g(0.6)=11.1\), \(g(0.8)=11.7\), \(g(1.0)=12.2\), \(g(1.2)=12.6\), \(g(1.4)=13.0\), \(g(1.6)=13.2\). (b) If \(-5 \leq g^{(4)}(x) \leq 2\) for \(0 \leq x \leq 1.6\), estimate the error involved in the approximation in part (a).
33 Exercise - Average Temperature · Level 2
A graph of the temperature in Boston on August 11, 2013, is shown. Use Simpson's Rule with \(n = 12\) to estimate the average temperature on that day.
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34 Exercise - Distance from Speed Data · Level 2
A radar gun was used to record the speed of a runner during the first \(5\) seconds of a race. Use Simpson's Rule to estimate the distance the runner covered during those \(5\) seconds. Data: \(v(0)=0\), \(v(0.5)=4.67\), \(v(1.0)=7.34\), \(v(1.5)=8.86\), \(v(2.0)=9.73\), \(v(2.5)=10.22\), \(v(3.0)=10.51\), \(v(3.5)=10.67\), \(v(4.0)=10.76\), \(v(4.5)=10.81\), \(v(5.0)=10.81\) (m/s).
35 Exercise - Velocity from Acceleration · Level 2
The graph of the acceleration \(a(t)\) of a car measured in ft/s² is shown. Use Simpson's Rule to estimate the increase in the velocity of the car during the \(6\)-second time interval.
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36 Exercise - Total Volume from Rate · Level 2
Water leaked from a tank at a rate of \(r(t)\) liters per hour, where the graph of \(r\) is as shown. Use Simpson's Rule to estimate the total amount of water that leaked out during the first \(6\) hours.
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37 Exercise - Energy from Power Data · Level 2
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 Simpson's Rule to estimate the energy used during that time period. (Use the fact that power is the derivative of energy.) Data: \(P(0:00)=1814\), \(P(0:30)=1735\), \(P(1:00)=1686\), \(P(1:30)=1646\), \(P(2:00)=1637\), \(P(2:30)=1609\), \(P(3:00)=1604\), \(P(3:30)=1611\), \(P(4:00)=1621\), \(P(4:30)=1666\), \(P(5:00)=1745\), \(P(5:30)=1886\), \(P(6:00)=2052\).
38 Exercise - Data Transmission · Level 2
Shown is the graph of traffic on an Internet service provider's T1 data line from midnight to 8:00 AM. \(D\) 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.
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39 Exercise - Volume of Revolution · Level 3
Use Simpson's 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.
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40 Exercise - Work from Force Data · Level 2
The table shows values of a force function \(f(x)\), where \(x\) is measured in meters and \(f(x)\) in newtons. Use Simpson's Rule to estimate the work done by the force in moving an object a distance of \(18\) m. Data: \(f(0)=9.8\), \(f(3)=9.1\), \(f(6)=8.5\), \(f(9)=8.0\), \(f(12)=7.7\), \(f(15)=7.5\), \(f(18)=7.4\).
41 Exercise - Volume of Solid · Level 3
The region bounded by the curve \(y = \dfrac{1}{1 + e^{-x}}\), the \(x\)- and \(y\)-axes, and the line \(x = 10\) is rotated about the \(x\)-axis. Use Simpson's Rule with \(n = 10\) to estimate the volume of the resulting solid.
42 Exercise - Pendulum Period · Level 3
The figure shows a pendulum with length \(L\) that makes a maximum angle \(\theta_0\) with the vertical. Using Newton's Second Law, it can be shown that the period \(T\) (the time for one complete swing) is given by \(T = 4 \sqrt{\dfrac{L}{g}} \displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{d x}{\sqrt{1 - k^2 \sin^2 x}}\) where \(k = \sin\left(\dfrac{1}{2} \theta_0\right)\) and \(g\) is the acceleration due to gravity. If \(L = 1\) m and \(\theta_0 = 42^{\circ}\), use Simpson's Rule with \(n = 10\) to find the period.
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43 Exercise - Diffraction Grating Intensity · Level 4
The intensity of light with wavelength \(\lambda\) traveling through a diffraction grating with \(N\) slits at an angle \(\theta\) is given by \(I(\theta) = N^2 \sin^2 k / k^2\), where \(k = (\pi N d \sin \theta)/\lambda\) and \(d\) is the distance between adjacent slits. A helium-neon laser with wavelength \(\lambda = 632.8 \times 10^{-9}\) m is emitting a narrow band of light, given by \(-10^{-6} < \theta < 10^{-6}\), through a grating with \$10,000\( slits spaced \)10^(-4)\( m apart. Use the Midpoint Rule with \)n = 10\( to estimate the total light intensity \)integral_(-10^(-6))^(10^(-6)) I(theta) d theta$ emerging from the grating.
44 Exercise - Trapezoidal Rule Periodic · Level 3
Use the Trapezoidal Rule with \(n = 10\) to approximate \(\displaystyle\int_{0}^{20} \cos(\pi x) d x\). Compare your result to the actual value. Can you explain the discrepancy?
45 Exercise - Conceptual Sketch · Level 3
Sketch the graph of a continuous function on \([0, 2]\) for which the Trapezoidal Rule with \(n = 2\) is more accurate than the Midpoint Rule.
46 Exercise - Conceptual Sketch · Level 3
Sketch the graph of a continuous function on \([0, 2]\) for which the right endpoint approximation with \(n = 2\) is more accurate than Simpson's Rule.
47 Exercise - Proof · Level 4
If \(f\) is a positive function and \(f''(x) < 0\) for \(a \leq x \leq b\), show that \(T_n < \displaystyle\int_{a}^{b} f(x) d x < M_n\).
48 Exercise - Proof · Level 4
Show that if \(f\) is a polynomial of degree \(3\) or lower, then Simpson's Rule gives the exact value of \(\displaystyle\int_{a}^{b} f(x) d x\).
49 Exercise - Proof · Level 4
Show that \(\dfrac{1}{2} (T_n + M_n) = T_{2 n}\).
50 Exercise - Proof · Level 4
Show that \(\dfrac{1}{3} T_n + \dfrac{2}{3} M_n = S_{2 n}\).
51 Example - Trapezoidal and Midpoint Rules · Level 2
Use (a) the Trapezoidal Rule and (b) the Midpoint Rule with \(n = 5\) to approximate the integral \(\displaystyle\int_{1}^{2} \dfrac{1}{x} d x\).
52 Example - Error Estimates · Level 3
How large should we take \(n\) in order to guarantee that the Trapezoidal and Midpoint Rule approximations for \(\displaystyle\int_{1}^{2} \dfrac{1}{x} d x\) are accurate to within \(0.0001\)?
53 Example - Midpoint Rule Application · Level 3
(a) Use the Midpoint Rule with \(n = 10\) to approximate the integral \(\displaystyle\int_{0}^{1} e^{x^2} d x\). (b) Give an upper bound for the error involved in this approximation.
54 Example - Simpson's Rule · Level 2
Use Simpson's Rule with \(n = 10\) to approximate \(\displaystyle\int_{1}^{2} \dfrac{1}{x} d x\).
55 Example - Simpson's Rule Data Application · Level 3
The figure shows data traffic on the link from the United States to SWITCH, the Swiss academic and research network, on February 10, 1998. \(D(t)\) is the data throughput, measured in megabits per second (Mb/s). Use Simpson's Rule to estimate the total amount of data transmitted on the link from midnight to noon on that day. Data values at hourly intervals: \(D(0)=3.2\), \(D(1)=2.7\), \(D(2)=1.9\), \(D(3)=1.7\), \(D(4)=1.3\), \(D(5)=1.0\), \(D(6)=1.1\), \(D(7)=1.3\), \(D(8)=2.8\), \(D(9)=5.7\), \(D(10)=7.1\), \(D(11)=7.7\), \(D(12)=7.9\) (Mb/s).
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56 Example - Error Estimate Simpson's Rule · Level 3
How large should we take \(n\) in order to guarantee that the Simpson's Rule approximation for \(\displaystyle\int_{1}^{2} \dfrac{1}{x} d x\) is accurate to within \(0.0001\)?
57 Example - Simpson's Rule with Error · Level 3
(a) Use Simpson's Rule with \(n = 10\) to approximate the integral \(\displaystyle\int_{0}^{1} e^{x^2} d x\). (b) Estimate the error involved in this approximation.

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