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# 7.6E: Exercises for Numerical Integration

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In exercises 1 - 5, approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)

1) $$\displaystyle ∫^2_1\frac{dx}{x};$$ trapezoidal rule; $$n=5$$

$$0.696$$

2) $$\displaystyle ∫^3_0\sqrt{4+x^3}\;dx;$$ trapezoidal rule; $$n=6$$

3) $$\displaystyle ∫^3_0\sqrt{4+x^3}\;dx;$$ Simpson’s rule; $$n=6$$

$$9.279$$

4) $$\displaystyle ∫^{12}_0x^2\;dx;$$ midpoint rule; $$n=6$$

5) $$\displaystyle ∫^1_0\sin^2(\pi x)\;dx;$$ midpoint rule; $$n=3$$

$$0.500$$

6) Use the midpoint rule with eight subdivisions to estimate $$\displaystyle ∫^4_2x^2\;dx.$$

7) Use the trapezoidal rule with four subdivisions to estimate $$\displaystyle ∫^4_2x^2\;dx.$$

$$T_4=18.75$$

8) Find the exact value of $$\displaystyle ∫^4_2x^2\;dx.$$ Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.

Approximate the integral to four decimal places using the indicated rule.

9) $$\displaystyle ∫^1_0\sin^2(\pi x)\;dx;$$ trapezoidal rule; $$n=6$$

$$0.5000$$

10) $$\displaystyle ∫^3_0\frac{1}{1+x^3}\;dx;$$ trapezoidal rule; $$n=6$$

11) $$\displaystyle ∫^3_0\frac{1}{1+x^3}\;dx;$$ Simpson’s rule; $$n=6$$

$$1.1614$$

12) $$\displaystyle ∫^{0.8}_0e^{−x^2}\;dx;$$ trapezoidal rule; $$n=4$$

13) $$\displaystyle ∫^{0.8}_0e^{−x^2}\;dx;$$ Simpson’s rule; $$n=4$$

$$0.6577$$

14) $$\displaystyle ∫^{0.4}_0\sin(x^2)\;dx;$$ trapezoidal rule; $$n=4$$

15) $$\displaystyle ∫^{0.4}_0\sin(x^2)\;dx;$$ Simpson’s rule; $$n=4$$

$$0.0213$$

16) $$\displaystyle ∫^{0.5}_{0.1}\frac{\cos x}{x}\;dx;$$ trapezoidal rule; $$n=4$$

17) $$\displaystyle ∫^{0.5}_{0.1}\frac{\cos x}{x}\;dx;$$ Simpson’s rule; $$n=4$$

$$1.5629$$

18) Evaluate $$\displaystyle ∫^1_0\frac{dx}{1+x^2}$$ exactly and show that the result is $$π/4$$. Then, find the approximate value of the integral using the trapezoidal rule with $$n=4$$ subdivisions. Use the result to approximate the value of $$π$$.

19) Approximate $$\displaystyle ∫^4_2\frac{1}{\ln x}\;dx$$ using the midpoint rule with four subdivisions to four decimal places.

$$1.9133$$

20) Approximate $$\displaystyle ∫^4_2\frac{1}{\ln x}\;dx$$ using the trapezoidal rule with eight subdivisions to four decimal places.

21) Use the trapezoidal rule with four subdivisions to estimate $$\displaystyle ∫^{0.8}_0x^3\;dx$$ to four decimal places.

$$T(4)=0.1088$$

22) Use the trapezoidal rule with four subdivisions to estimate $$\displaystyle ∫^{0.8}_0x^3\;dx.$$ Compare this value with the exact value and find the error estimate.

23) Using Simpson’s rule with four subdivisions, find $$\displaystyle ∫^{π/2}_0\cos(x)\;dx.$$

$$\displaystyle ∫^{π/2}_0\cos(x)\;dx\approx \quad 1.0$$

24) Show that the exact value of $$\displaystyle ∫^1_0xe^{−x}\;dx=1−\frac{2}{e}$$. Find the absolute error if you approximate the integral using the midpoint rule with 16 subdivisions.

25) Given $$\displaystyle ∫^1_0xe^{−x}\;dx=1−\frac{2}{e},$$ use the trapezoidal rule with 16 subdivisions to approximate the integral and find the absolute error.

Approximate error is $$0.000325.$$

26) Find an upper bound for the error in estimating $$\displaystyle ∫^3_0(5x+4)\;dx$$ using the trapezoidal rule with six steps.

27) Find an upper bound for the error in estimating $$\displaystyle ∫^5_4\frac{1}{(x−1)^2}\;dx$$ using the trapezoidal rule with seven subdivisions.

$$\frac{1}{7938}$$

28) Find an upper bound for the error in estimating $$\displaystyle ∫^3_0(6x^2−1)\;dx$$ using Simpson’s rule with $$n=10$$ steps.

29) Find an upper bound for the error in estimating $$\displaystyle ∫^5_2\frac{1}{x−1}\;dx$$ using Simpson’s rule with $$n=10$$ steps.

$$\frac{81}{25,000}$$

30) Find an upper bound for the error in estimating $$\displaystyle ∫^π_02x\cos(x)\;dx$$ using Simpson’s rule with four steps.

31) Estimate the minimum number of subintervals needed to approximate the integral $$\displaystyle ∫^4_1(5x^2+8)\;dx$$ with an error magnitude of less than 0.0001 using the trapezoidal rule.

$$475$$

32) Determine a value of n such that the trapezoidal rule will approximate $$\displaystyle ∫^1_0\sqrt{1+x^2}\;dx$$ with an error of no more than 0.01.

33) Estimate the minimum number of subintervals needed to approximate the integral $$\displaystyle ∫^3_2(2x^3+4x)\;dx$$ with an error of magnitude less than 0.0001 using the trapezoidal rule.

$$174$$

34) Estimate the minimum number of subintervals needed to approximate the integral $$\displaystyle ∫^4_3\frac{1}{(x−1)^2}\;dx$$ with an error magnitude of less than 0.0001 using the trapezoidal rule.

35) Use Simpson’s rule with four subdivisions to approximate the area under the probability density function $$y=\frac{1}{\sqrt{2π}}e^{−x^2/2}$$ from $$x=0$$ to $$x=0.4$$.

$$0.1544$$

36) Use Simpson’s rule with $$n=14$$ to approximate (to three decimal places) the area of the region bounded by the graphs of $$y=0, x=0,$$ and $$x=π/2.$$

37) The length of one arch of the curve $$y=3\sin(2x)$$ is given by $$L=∫^{π/2}_0\sqrt{1+36\cos^2(2x)}\;dx.$$ Estimate L using the trapezoidal rule with $$n=6$$.

$$6.2807$$

38) The length of the ellipse $$x=a\cos(t),y=b\sin(t),0≤t≤2π$$ is given by $$L=4a∫^{π/2}_0\sqrt{1−e^2\cos^2(t)}dt$$, where e is the eccentricity of the ellipse. Use Simpson’s rule with $$n=6$$ subdivisions to estimate the length of the ellipse when $$a=2$$ and $$e=1/3.$$

39) Estimate the area of the surface generated by revolving the curve $$y=\cos(2x),0≤x≤\frac{π}{4}$$ about the x-axis. Use the trapezoidal rule with six subdivisions.

$$4.606$$

40) Estimate the area of the surface generated by revolving the curve $$y=2x^2, 0≤x≤3$$ about the x-axis. Use Simpson’s rule with $$n=6.$$

41) The growth rate of a certain tree (in feet) is given by $$y=\dfrac{2}{t+1}+e^{−t^2/2},$$ where t is time in years. Estimate the growth of the tree through the end of the second year by using Simpson’s rule, using two subintervals. (Round the answer to the nearest hundredth.)

$$3.41$$ ft

42) [T] Use a calculator to approximate $$\displaystyle ∫^1_0\sin(πx)\;dx$$ using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.

43) [T] Given $$\displaystyle ∫^5_1(3x^2−2x)\;dx=100,$$ approximate the value of this integral using the midpoint rule with 16 subdivisions and determine the absolute error.

$$T_{16}=100.125;$$ absolute error = $$0.125$$

44) Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?

45) The table represents the coordinates $$(x,​y)$$ that give the boundary of a lot. The units of measurement are meters. Use the trapezoidal rule to estimate the number of square meters of land that is in this lot.

 $$x$$ $$y$$ $$x$$ $$y$$ 0 125 600 95 100 125 700 88 200 120 800 75 300 112 900 35 400 90 1000 0 500 90

46) Choose the correct answer. When Simpson’s rule is used to approximate the definite integral, it is necessary that the number of partitions be____

a. an even number

b. odd number

c. either an even or an odd number

d. a multiple of 4

47) The “Simpson” sum is based on the area under a ____.

a. $$f(x)$$
b. $$f′(x)$$
c. $$f^{(4)}(x)$$