
2.5E: Exercises


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; $$\displaystyle n=5$$

$$\displaystyle 0.696$$

Exercise $$\PageIndex{2}$$

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

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Exercise $$\PageIndex{3}$$

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

$$\displaystyle 9.279$$

Exercise $$\PageIndex{4}$$

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

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Exercise $$\PageIndex{5}$$

$$\displaystyle ∫^1_0sin^2(πx)dx;$$ midpoint rule; $$\displaystyle n=3$$

$$\displaystyle 0.5000$$

Exercise $$\PageIndex{6}$$

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

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Exercise $$\PageIndex{7}$$

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

$$\displaystyle T_4=18.75$$

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.

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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; $$\displaystyle n=6$$

$$\displaystyle 0.500$$

Exercise $$\PageIndex{10}$$

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

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Exercise $$\PageIndex{11}$$

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

$$\displaystyle 1.1614$$

Exercise $$\PageIndex{12}$$

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

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Exercise $$\PageIndex{13}$$

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

$$\displaystyle 0.6577$$

Exercise $$\PageIndex{14}$$

$$\displaystyle ∫^{0.4}_0sin(x^2)dx;$$ trapezoidal rule; $$\displaystyle n=4$$

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Exercise $$\PageIndex{15}$$

$$\displaystyle ∫^{0.4}_0sin(x^2)dx;$$ Simpson’s rule; $$\displaystyle n=4$$

$$\displaystyle 0.0213$$

Exercise $$\PageIndex{16}$$

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

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Exercise $$\PageIndex{17}$$

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

$$\displaystyle 1.5629$$

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 $$\displaystyle π$$.

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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.

$$\displaystyle 1.9133$$

Exercise $$\PageIndex{20}$$

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

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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.

$$\displaystyle T(4)=0.1088$$

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.

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Exercise $$\PageIndex{23}$$

Using Simpson’s rule with four subdivisions, find $$\displaystyle ∫^{π/2}_0cos(x)dx.$$

$$\displaystyle 1.0$$

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.

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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.

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.

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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.

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

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.

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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.

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

Exercise $$\PageIndex{30}$$

Find an upper bound for the error in estimating $$\displaystyle ∫^π_02xcos(x)dx$$ using Simpson’s rule with four steps.

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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.

$$\displaystyle 475$$

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.

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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.

$$\displaystyle 174$$

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.

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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 $$\displaystyle x=0.4$$.

$$\displaystyle 0.1544$$

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 $$\displaystyle x=π/2.$$

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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 $$\displaystyle n=6$$.

$$\displaystyle 6.2807$$

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 $$\displaystyle e=1/3.$$

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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.

$$\displaystyle 4.606$$

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 $$\displaystyle n=6.$$

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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.)

$$\displaystyle 3.41$$ 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.

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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.

$$\displaystyle T_{16}=100.125;$$ 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?

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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

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

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Exercise $$\PageIndex{47}$$

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

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