Stewart 8th Section 7.7: Approximate Integration

1 Integration and Accumulation of Change · 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.

2 Integration and Accumulation of Change · 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?

3 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Differentiation: Composite, Implicit, and Inverse Functions · 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 Infinite Sequences and Series · 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 Infinite Sequences and Series · 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 Infinite Sequences and Series · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · Level 4

Repeat Exercise 23 for the integral $\displaystyle\int_{-1}^1 \sqrt{4 - x^3} d x$.

25 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 $n = 6$.

30 Integration and Accumulation of Change · 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.

31 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Applications of Integration · 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.

34 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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.

36 Integration and Accumulation of Change · 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.

37 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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.

39 Applications of Integration · 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.

40 Applications of Integration · 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 Applications of Integration · 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 Contextual Applications of Differentiation · 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.

43 Analytical Applications of Differentiation · 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 Integration and Accumulation of Change · 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 Analytical Applications of Differentiation · 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 Analytical Applications of Differentiation · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · 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 Differentiation: Definition and Fundamental Properties · Level 4

Show that $\dfrac{1}{2} (T_n + M_n) = T_{2 n}$.

50 Differentiation: Definition and Fundamental Properties · Level 4

Show that $\dfrac{1}{3} T_n + \dfrac{2}{3} M_n = S_{2 n}$.

51 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · Level 2

Use Simpson's Rule with $n = 10$ to approximate $\displaystyle\int_{1}^{2} \dfrac{1}{x} d x$.

55 Integration and Accumulation of Change · 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).

56 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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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