Final Exam: Review Guide
This page contains problems similar to those that will appear on the
Final Exam.
You should study the midterm exam review guide as well as the following
problems.
1. Compute \(\displaystyle \int_0^2 x^2 - x\, dx\) by using the Riemann
sum definition of the integral.
2. Evaluate the integral by interpreting it in terms of areas.
\[ \int_0^1 \Big(x + \sqrt{1 - x^2}\Big)\, dx \]
3. If \( \int_0^6 f(x)\, dx = 10\) and \( \int_0^4 f(x)\, dx = 7\), find
\( \int_4^6 f(x)\, dx\).
4. Evaluate:
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a.) |
\(\displaystyle \int_0^{\pi/2} \frac{d}{dx}\left(\sin\frac{x}{2}
\cos\frac{x}{3}\right)\, dx\)
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b.) |
\(\displaystyle \frac{d}{dx} \int_0^{\pi/2} \sin\frac{x}{2}
\cos\frac{x}{3}\, dx\)
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c.) |
\(\displaystyle \frac{d}{dx} \int_x^{\pi/2} \sin\frac{t}{2}
\cos\frac{t}{3}\, dt\) |
5. Compute \(\displaystyle \int \dfrac{(\arctan x)^2}{1 + x^2} \, dx\).
6. a.) If \(f\) is continuous on \([0,\pi]\), use the substitution
\(u = \pi - x\) to show that
\[ \int_0^\pi x\,f(\sin x)\, dx = \dfrac{\pi}{2} \int_0^\pi f(\sin x)\, dx.\]
b.) Evaluate \(\displaystyle \int_0^\pi \dfrac{x\, \sin x}{1 + \cos^2 x}\, dx\).
7. A particle moves along a line with velocity function \(v(t) = t^2 - t\),
where \(v\) is measured in meters per second. Find the displacement and the distance
traveled by the particle during the time interval \([0,5]\).
8. Let \(r(t)\) be the rate at which the world's oil is consumed, where \(t\)
is measured in years starting at \(t = 0\) on January 1, 2000, and \(r(t)\) is
measured in barrels per year. What does \(\int_0^8 r(t)\, dt\) represent?
9. If \(f\) is continuous and \(\int_0^2 f(x)\, dx = 6\), evaluate
\(\int_0^{\pi/2} f(2\sin\theta)\cos\theta\,d\theta\).
10. If \(f\) is a continuous function such that
\[ \int_0^x f(t)\, dt = x\sin x + \int_0^x \dfrac{f(t)}{1 + t^2}\, dt \]
for all \(x\), find and explicit formula for \(f(x)\).
11. If \(f\) is continuous on \([0,1]\), prove that
\[ \int_0^1 f(x)\, dx = \int_0^1 f(1-x)\, dx.\]
12. Find the area of the region bounded by the curves \(y = \sin(\pi x/2)\)
and \(y = x^2 - 2x\).
13. Find the volume of the solid obtained by rotating the region bounded by
the curves about the line \(y = -1\).
\[ y = x^2 + 1,\ \ \ y = 9 - x^2\]
14. Find the volumes of the solids obtained by rotating the region bounded by
the curves \(y = x\) and \(y = x^2\) about the following lines.
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a.) |
the \(x\)-axis
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b.) |
the \(y\)-axis
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c.) |
\(y = 2\)
|
15. The following integral represents the volume of a solid of revolution.
Describe the solid.
\[ \int_0^4 2\pi(6-y)(4y - y^2)\, dy \]
16. A force of 30 N is required to maintain a spring stretched from its
natural length of 12 cm to a length of 15 cm. How much work is done in stretching
the spring from 12 cm to 20 cm?
17. Find the average value of the function \(f(t) = \sec^2(t)\) on the
interval \([0,\pi/4]\).
18. If \(f\) is a continuous function, what is the limit as \(h \to 0\)
of the average value of \(f\) on the interval \([x,x+h]\)?
19. Show that
\[\frac{d}{dx}\left(\frac{1}{2}\arctan(x) + \frac{1}{4}
\ln\left(\frac{(x+1)^2}{x^2 + 1}\right)\right) = \dfrac{1}{(1+x)(1+x^2)}\]
20. At what point on the curve \(y = \left[ \ln(x+4) \right]^2\)
is the tangent line horizontal?
21. If \(f(x) = x + x^2 + e^x\), find \((f^{-1})'(1)\).
22. Show that
\[\cos\left(\arctan\left(\sin\left({\rm arccot}\, x\right) \right) \right) =
\sqrt{\dfrac{x^2 + 1}{x^2 + 2}}.\]
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