Warm-Up.
Answer each of the following as best you can. Feel free to discuss your thoughts with peers around you.
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Worksheet Session #11: Final Exam Review Part 1 -- Chapters 6 & 7
Directions: Choose the integration technique that can be used to solve each of the following integrals. You do not need to solve the integrals; just identify the method.
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Integration by Parts
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Partial Fractions
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Trigonometric Substitution (e.g., \(x = a\sin(\theta)\text{,}\) etc.)
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\(u\)-substitution
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Integration using Trigonometric Identities
1.
\(\displaystyle\int \frac{1}{4 + x^2}\,dx\)
2.
\(\displaystyle\int \frac{10}{(x-1)(x^2+9)}\,dx\)
3.
\(\displaystyle\int x^2 e^{-x}\,dx\)
4.
\(\displaystyle\int \frac{1}{16}x^3 \sin\!\left(x^4\right)\,dx\)
5.
\(\displaystyle\int \sin^4(x)\,dx\)
Directions: Answer each of the following with “true” or “false”.
6.
\(\displaystyle\int_{-1}^{1} \frac{1}{x^2}\,dx\) is an improper integral.
7.
\(\displaystyle\int_{1}^{\infty} \frac{1}{x^{2p}}\,dx\) converges for \(p \geq \dfrac{1}{2}\text{.}\)
8.
The series \(\displaystyle 7 - \frac{14}{3} + \frac{28}{9} - \frac{56}{27} + \frac{112}{81} - \frac{224}{243} + \cdots\) converges.
Hands-On.
Answer each of the following as best you can. Feel free to discuss your thoughts with peers around you.
1.
Let \(R\) denote the region bounded by \(y = e^x\) and \(y = e^{-x}\) between \(x = -1\) and \(x = 2\text{.}\)
(a)
Sketch the region \(R\text{.}\)
(b)
Set up the integral that gives the area of the region \(R\text{.}\)
(c)
Find the area of the region \(R\text{.}\)
2.
Suppose the work required to stretch a spring 4 meters beyond its natural length is 24 J. How much force would be required to hold the spring stretched 10 meters beyond its natural length?
3.
A 20 ft rope weighing 3 lb/ft hangs vertically from the edge of a roof. A 10 lb bucket is attached to the lower end of the rope. Find the work required to pull the rope and bucket to the roof.
4.
Let \(R\) denote the region bounded by \(y = \ln(x)\text{,}\) \(y = 0\text{,}\) and \(x = 2\text{.}\)
(a)
Sketch the region \(R\text{.}\)
(b)
Use disks to setup the integral that gives the volume of the solid generated by rotating the region \(R\) about the \(x\)-axis. Do not evaluate the integral.
(c)
Use washers to setup the integral that gives the volume of the solid generated by rotating the region \(R\) about the line \(x = 3\text{.}\) Do not evaluate the integral.
(d)
Use shells to setup the integral that gives the volume of the solid generated by rotating the region \(R\) about the line \(y = -2\text{.}\) Do not evaluate the integral.
5.
The tank in a shape of a hemisphere with radius 5 ft is full of water. Set up, but do not evaluate, the integral that gives the work done in pumping all the water to the top of the tank. Use the fact that water weighs 62.5 lb/ft\(^3\text{.}\)
6.
The base of a solid is the region bounded by the curve \(y = 5 - x^2\) and the \(x\)-axis. The cross sections of the solid are rectangles perpendicular to the \(y\)-axis with height equal to twice the base. Find the volume of this solid.
7.
Compute \(\displaystyle\int x^2 e^{3x}\,dx\)
8.
Compute \(\displaystyle\int \sin^6(x)\cos^3(x)\,dx\)
9.
Compute \(\displaystyle\int \frac{2x^3 + 11x^2 + 18}{x^2(x^2 + 9)}\,dx\)
10.
Compute \(\displaystyle\int \frac{1}{(4 + 9x^2)^{5/2}}\,dx\)
11.
Compute \(\displaystyle\int_{3}^{\infty} \frac{1}{x\,\left(\ln(x)\right)^2}\,dx\)
12.
Compute \(\displaystyle\int_{0}^{1} \frac{x + 4}{x^2 + 1}\,dx\)
Grades Up!
Choose the best answer for each of the following.
1.
A certain substitution gives \(\displaystyle\int \sin^3(x)\cos^n(x)\,dx = \int u^9 - u^7\,du\text{.}\) Given this, \(n\) must be .
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\(\displaystyle 1\)
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\(\displaystyle 3\)
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\(\displaystyle 5\)
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\(\displaystyle 7\)
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None of these.
2.
Suppose \(f\) is a function such that \(f'\) and \(f''\) exist, and \(f(0) = 1\text{,}\) \(f'(0) = 5\text{,}\) \(f(2) = 3\text{,}\) \(f'(2) = 7\text{.}\) Which of the following correctly evaluates \(\displaystyle\int_{0}^{2} 3x f''(x)\,dx\text{?}\)
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\(\displaystyle 36\)
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\(\displaystyle 32\)
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\(\displaystyle 28\)
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\(\displaystyle 24\)
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\(\displaystyle 22\)
3.
The base of a solid is the triangle enclosed by \(x + y = 5\text{,}\) the \(x\)-axis, and the \(y\)-axis. Its cross sections are semicircles perpendicular to the \(y\)-axis. Which integral below calculates the volume of the solid?
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\(\displaystyle \displaystyle\frac{\pi}{2}\int_{0}^{5}(5-y)^2\,dy\)
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\(\displaystyle \displaystyle\frac{\pi}{8}\int_{0}^{5}(5-y)^2\,dy\)
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\(\displaystyle \displaystyle\frac{\pi}{4}\int_{0}^{5}(5-x)^2\,dx\)
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\(\displaystyle \displaystyle\int_{0}^{5}(5-y)^2\,dy\)
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\(\displaystyle \displaystyle 4\int_{-2}^{2}(4-y^2)\,dx\)
4.
Which of the following integrals gives the volume of the solid obtained by revolving the region bounded by \(y = 4x - x^2\) and \(y = 0\) about the line \(x = -2\text{?}\)
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\(\displaystyle \displaystyle 2\pi\int_0^4 (2-x)(4x-x^2)\,dx\)
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\(\displaystyle \displaystyle 2\pi\int_0^4 x(4x-x^2)\,dx\)
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\(\displaystyle \displaystyle 2\pi\int_0^4 (x+2)(4x-x^2)\,dx\)
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\(\displaystyle \displaystyle 2\pi\int_0^4 (x-2)(4x-x^2)\,dx\)
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None of these.
