2.5E: Exercises
- Page ID
- 18586
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Approximate the following integrals using either the midpoint rule, trapezoidal rule, or Simpson’s rule as indicated. (Round answers to three decimal places.)
Exercise \(\PageIndex{1}\)
\(\displaystyle ∫^2_1\frac{dx}{x};\) trapezoidal rule; \(\displaystyle n=5\)
- Answer
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\(\displaystyle 0.696\)
Exercise \(\PageIndex{2}\)
\(\displaystyle ∫^3_0\sqrt{4+x^3}dx;\) trapezoidal rule; \(\displaystyle n=6\)
- Answer
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Exercise \(\PageIndex{3}\)
\(\displaystyle ∫^3_0\sqrt{4+x^3}dx;\) Simpson’s rule; \(\displaystyle n=3\)
- Answer
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\(\displaystyle 9.279\)
Exercise \(\PageIndex{4}\)
\(\displaystyle ∫^{12}_0x^2dx;\) midpoint rule; \(\displaystyle n=6\)
- Answer
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Exercise \(\PageIndex{5}\)
\(\displaystyle ∫^1_0sin^2(πx)dx;\) midpoint rule; \(\displaystyle n=3\)
- Answer
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\(\displaystyle 0.5000\)
Exercise \(\PageIndex{6}\)
Use the midpoint rule with eight subdivisions to estimate \(\displaystyle ∫^4_2x^2dx.\)
- Answer
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Exercise \(\PageIndex{7}\)
Use the trapezoidal rule with four subdivisions to estimate \(\displaystyle ∫^4_2x^2dx.\)
- Answer
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\(\displaystyle T_4=18.75\)
Exercise \(\PageIndex{8}\)
Find the exact value of \(\displaystyle ∫^4_2x^2dx.\) 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.
- Answer
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Approximate the integral to three decimal places using the indicated rule.
Exercise \(\PageIndex{9}\)
\(\displaystyle ∫^1_0sin^2(πx)dx;\) trapezoidal rule; \(\displaystyle n=6\)
- Answer
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\(\displaystyle 0.500\)
Exercise \(\PageIndex{10}\)
\(\displaystyle ∫^3_0\frac{1}{1+x^3}dx;\) trapezoidal rule; \(\displaystyle n=6\)
- Answer
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Exercise \(\PageIndex{11}\)
\(\displaystyle ∫^3_0\frac{1}{1+x^3}dx;\) Simpson’s rule; \(\displaystyle n=3\)
- Answer
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\(\displaystyle 1.1614\)
Exercise \(\PageIndex{12}\)
\(\displaystyle ∫^{0.8}_0e^{−x^2}dx;\) trapezoidal rule; \(\displaystyle n=4\)
- Answer
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Exercise \(\PageIndex{13}\)
\(\displaystyle ∫^{0.8}_0e^{−x^2}dx;\) Simpson’s rule; \(\displaystyle n=4\)
- Answer
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\(\displaystyle 0.6577\)
Exercise \(\PageIndex{14}\)
\(\displaystyle ∫^{0.4}_0sin(x^2)dx;\) trapezoidal rule; \(\displaystyle n=4\)
- Answer
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Exercise \(\PageIndex{15}\)
\(\displaystyle ∫^{0.4}_0sin(x^2)dx;\) Simpson’s rule; \(\displaystyle n=4\)
- Answer
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\(\displaystyle 0.0213\)
Exercise \(\PageIndex{16}\)
\(\displaystyle ∫^{0.5}_{0.1}\frac{cosx}{x}dx;\) trapezoidal rule; \(\displaystyle n=4\)
- Answer
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Exercise \(\PageIndex{17}\)
\(\displaystyle ∫^{0.5}_{0.1}\frac{cosx}{x}dx;\) Simpson’s rule; \(\displaystyle n=4\)
- Answer
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\(\displaystyle 1.5629\)
Exercise \(\PageIndex{18}\)
Evaluate \(\displaystyle ∫^1_0\frac{dx}{1+x^2}\) exactly and show that the result is \(\displaystyle π/4\). Then, find the approximate value of the integral using the trapezoidal rule with \(\displaystyle n=4\) subdivisions. Use the result to approximate the value of \(\displaystyle π\).
- Answer
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Exercise \(\PageIndex{19}\)
Approximate \(\displaystyle ∫^4_2\frac{1}{lnx}dx\) using the midpoint rule with four subdivisions to four decimal places.
- Answer
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\(\displaystyle 1.9133\)
Exercise \(\PageIndex{20}\)
Approximate \(\displaystyle ∫^4_2\frac{1}{lnx}dx\) using the trapezoidal rule with eight subdivisions to four decimal places.
- Answer
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Exercise \(\PageIndex{21}\)
Use the trapezoidal rule with four subdivisions to estimate \(\displaystyle ∫^{0.8}_0x^3dx\) to four decimal places.
- Answer
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\(\displaystyle T(4)=0.1088\)
Exercise \(\PageIndex{22}\)
Use the trapezoidal rule with four subdivisions to estimate \(\displaystyle ∫^{0.8}_0x^3dx.\) Compare this value with the exact value and find the error estimate.
- Answer
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Exercise \(\PageIndex{23}\)
Using Simpson’s rule with four subdivisions, find \(\displaystyle ∫^{π/2}_0cos(x)dx.\)
- Answer
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\(\displaystyle 1.0\)
Exercise \(\PageIndex{24}\)
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.
- Answer
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Exercise \(\PageIndex{25}\)
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
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Approximate error is \(\displaystyle 0.000325.\)
Exercise \(\PageIndex{26}\)
Find an upper bound for the error in estimating \(\displaystyle ∫^3_0(5x+4)dx\) using the trapezoidal rule with six steps.
- Answer
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Exercise \(\PageIndex{27}\)
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
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\(\displaystyle \frac{1}{7938}\)
Exercise \(\PageIndex{28}\)
Find an upper bound for the error in estimating \(\displaystyle ∫^3_0(6x^2−1)dx\) using Simpson’s rule with \(\displaystyle n=10\) steps.
- Answer
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Exercise \(\PageIndex{29}\)
Find an upper bound for the error in estimating \(\displaystyle ∫^5_2\frac{1}{x−1}dx\) using Simpson’s rule with \(\displaystyle n=10\) steps.
- Answer
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\(\displaystyle \frac{81}{25,000}\)
Exercise \(\PageIndex{30}\)
Find an upper bound for the error in estimating \(\displaystyle ∫^π_02xcos(x)dx\) using Simpson’s rule with four steps.
- Answer
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Exercise \(\PageIndex{31}\)
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
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\(\displaystyle 475\)
Exercise \(\PageIndex{32}\)
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.
- Answer
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Exercise \(\PageIndex{33}\)
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
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\(\displaystyle 174\)
Exercise \(\PageIndex{34}\)
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.
- Answer
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Exercise \(\PageIndex{35}\)
Use Simpson’s rule with four subdivisions to approximate the area under the probability density function \(\displaystyle y=\frac{1}{\sqrt{2π}}e^{−x^2/2}\) from \(\displaystyle x=0\) to \(\displaystyle x=0.4\).
- Answer
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\(\displaystyle 0.1544\)
Exercise \(\PageIndex{36}\)
Use Simpson’s rule with \(\displaystyle n=14\) to approximate (to three decimal places) the area of the region bounded by the graphs of \(\displaystyle y=0, x=0,\) and \(\displaystyle x=π/2.\)
- Answer
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Exercise \(\PageIndex{37}\)
The length of one arch of the curve \(\displaystyle y=3sin(2x)\) is given by \(\displaystyle L=∫^{π/2}_0\sqrt{1+36cos^2(2x)}dx.\) Estimate L using the trapezoidal rule with \(\displaystyle n=6\).
- Answer
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\(\displaystyle 6.2807\)
Exercise \(\PageIndex{38}\)
The length of the ellipse \(\displaystyle x=acos(t),y=bsin(t),0≤t≤2π\) is given by \(\displaystyle L=4a∫^{π/2}_0\sqrt{1−e^2cos^2(t)}dt\), where e is the eccentricity of the ellipse. Use Simpson’s rule with \(\displaystyle n=6\) subdivisions to estimate the length of the ellipse when \(\displaystyle a=2\) and \(\displaystyle e=1/3.\)
- Answer
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Exercise \(\PageIndex{39}\)
Estimate the area of the surface generated by revolving the curve \(\displaystyle y=cos(2x),0≤x≤\frac{π}{4}\) about the x-axis. Use the trapezoidal rule with six subdivisions.
- Answer
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\(\displaystyle 4.606\)
Exercise \(\PageIndex{40}\)
Estimate the area of the surface generated by revolving the curve \(\displaystyle y=2x^2, 0≤x≤3\) about the x-axis. Use Simpson’s rule with \(\displaystyle n=6.\)
- Answer
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Exercise \(\PageIndex{41}\)
The growth rate of a certain tree (in feet) is given by \(\displaystyle y=\frac{2}{t+1}+e^{−t^2/2},\) 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
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\(\displaystyle 3.41\) ft
Exercise \(\PageIndex{42}\)
Use a calculator to approximate \(\displaystyle ∫^1_0sin(πx)dx\) using the midpoint rule with 25 subdivisions. Compute the relative error of approximation.
- Answer
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Exercise \(\PageIndex{43}\)
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
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\(\displaystyle T_{16}=100.125;\) absolute error = \(\displaystyle 0.125\)
Exercise \(\PageIndex{44}\)
Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?
- Answer
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Exercise \(\PageIndex{45}\)
The table represents the coordinates \(\displaystyle (x,y)\) 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.
| \(\displaystyle x\) | \(\displaystyle y\) | \(\displaystyle x\) | \(\displaystyle y\) |
| 0 | 125 | 600 | 95 |
| 100 | 125 | 700 | 88 |
| 200 | 120 | 800 | 75 |
| 300 | 112 | 900 | 35 |
| 400 | 90 | 1000 | 0 |
| 500 | 90 |
- Answer
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about 89,250 m2
Exercise \(\PageIndex{46}\)
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
- Answer
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Exercise \(\PageIndex{47}\)
The “Simpson” sum is based on the area under a ____.
- Answer
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parabola
Exercise \(\PageIndex{48}\)
The error formula for Simpson’s rule depends on___.
a. \(\displaystyle f(x)\)
b. \(\displaystyle f′(x)\)
c. \(\displaystyle f^{(4)}(x)\)
d. the number of steps
- Answer
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