7.6E: Exercises for Section 7.6

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; \( n=5\)

Answer:

\( 0.696\)

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; \( n=6\)

Answer:

\( 9.279\)

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

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

Answer:

\( 0.500\)

 

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 \( \displaystyle ∫^4_2x^2\;dx.\)

Answer:

\( T_4=18.75\)

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; \( n=6\)

Answer:

\( 0.5000\)

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; \( n=6\)

Answer:

\( 1.1614\)

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

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

Answer:

\(0.6577\)

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; \( n=4\)

Answer:

\(0.0213\)

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; \(n=4\)

Answer:

\(1.5629\)

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.

Answer:

\( 1.9133\)

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.

Answer:

\( T(4)=0.1088\)

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 \( \displaystyle ∫^{π/2}_0\cos(x)\;dx.\)

Answer:

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

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.

Answer:

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.

Answer:

\( \frac{1}{7938}\)

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.

Answer:

\( \frac{81}{25,000}\)

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.

Answer:

\( 475\)

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.

Answer:

\( 174\)

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 \( x=0.4\).

Answer:

\( 0.1544\)

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 \( n=6\).

Answer:

\( 6.2807\)

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.

Answer:

\( 4.606\)

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.)

Answer:

\( 3.41\) 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.

Answer:

\( T_{16}=100.125;\) 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

Answer:

about 89,250 m²

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 ____.

Answer:

parabola

48) The error formula for Simpson’s rule depends on___.

a. \( f(x)\)

b. \( f′(x)\)

c. \( f^{(4)}(x)\)

d. the number of steps