# 2.5E: Exercises

- Page ID
- 18586

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**-
\(\displaystyle 0.696\)

## 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**-
\(\displaystyle 9.279\)

## 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**-
\(\displaystyle 0.5000\)

## 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**-
\(\displaystyle T_4=18.75\)

## 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**-
\(\displaystyle 0.500\)

## 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**-
\(\displaystyle 1.1614\)

## 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**-
\(\displaystyle 0.6577\)

## 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**-
\(\displaystyle 0.0213\)

## 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**-
\(\displaystyle 1.5629\)

## 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**-
\(\displaystyle 1.9133\)

## 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**-
\(\displaystyle T(4)=0.1088\)

## 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**-
\(\displaystyle 1.0\)

## 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**-
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**-
\(\displaystyle \frac{1}{7938}\)

## 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**-
\(\displaystyle \frac{81}{25,000}\)

## 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**-
\(\displaystyle 475\)

## 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**-
\(\displaystyle 174\)

## 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**-
\(\displaystyle 0.1544\)

## 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**-
\(\displaystyle 6.2807\)

## 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**-
\(\displaystyle 4.606\)

## 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**-
\(\displaystyle 3.41\) 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**-
\(\displaystyle T_{16}=100.125;\) 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**-
about 89,250 m

^{2}

## 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**-
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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