Homework 10: Calculus and Analytic Geometry

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 24 February 2010

Problem A. Integrals with exponential functions and logarithms.

1. $\int e^{2x-1}\mathrm{dx}$
2. $\int e^{1-3x}\mathrm{dx}$
3. $\int 3^{2-3x}\mathrm{dx}$
4. $\int \frac{1}{x\ln x}\mathrm{dx}$
5. $\int \frac{\ln \left(x^2\right)}{x}\mathrm{dx}$
6. $\int \frac{{\left(\ln \left(x\right)\right)}^2}{x}\mathrm{dx}$
7. $\int x^{-2}{e}^{-\frac{1}{x}}\mathrm{dx}$
8. $\int \frac{e^x}{1+e^{2x}}\mathrm{dx}$
9. $\int \frac{e^{2x}}{e^{2x}-2}\mathrm{dx}$
10. $\int \frac{\sqrt{2+\ln x}}{x}\mathrm{dx}$
11. $\int \frac{\left(x+1\right){\left(x+\ln x\right)}^2}{x}\mathrm{dx}$
12. $\int \sqrt{e^x-1}\mathrm{dx}$

Problem B. Definite integrals.

1. $\underset{-2}{\overset{4}{\int }}\left(3x-5\right)\mathrm{dx}$
2. $\underset{1}{\overset{2}{\int }}{x}^{-2}\mathrm{dx}$
3. $\underset{0}{\overset{1}{\int }}\left(1-2x-3x^2\right)\mathrm{dx}$
4. $\underset{1}{\overset{2}{\int }}\left(5x^2-4x+3\right)\mathrm{dx}$
5. $\underset{-3}{\overset{0}{\int }}\left(5y^4-6y^2+14\right)\mathrm{dy}$
6. $\underset{0}{\overset{1}{\int }}\left(y^9-2y^5+3y\right)\mathrm{dy}$
7. $\underset{0}{\overset{4}{\int }}\sqrt{x}\mathrm{dx}$
8. $\underset{0}{\overset{1}{\int }}{x}^{\frac{3}{7}}\mathrm{dx}$
9. $\underset{1}{\overset{3}{\int }}\left(\frac{1}{t^2}-\frac{1}{t^4}\right)\mathrm{dt}$
10. $\underset{2}{\overset{1}{\int }}\frac{t^6-t^2}{t^4}\mathrm{dt}$
11. $\underset{1}{\overset{2}{\int }}{\left(x^3-1\right)}^2\mathrm{dx}$
12. $\underset{1}{\overset{2}{\int }}\frac{x^2+1}{\sqrt{x}}\mathrm{dx}$
13. $\underset{0}{\overset{1}{\int }}u\left(\sqrt{u}+\sqrt[3]{u}\right)\mathrm{du}$
14. $\underset{-1}{\overset{1}{\int }}\frac{3}{t^4}\mathrm{dt}$
15. $\underset{1}{\overset{2}{\int }}{\left(x+\frac{1}{x}\right)}^2\mathrm{dx}$
16. $\underset{3}{\overset{3}{\int }}\sqrt{x^5+2}\mathrm{dx}$
17. $\underset{-4}{\overset{2}{\int }}\frac{2}{x^6}\mathrm{dx}$
18. $\underset{1}{\overset{-1}{\int }}\left(x-1\right)\left(3x+2\right)\mathrm{dx}$
19. $\underset{1}{\overset{4}{\int }}\left(\sqrt{t}-\frac{2}{\sqrt{t}}\right)\mathrm{dt}$
20. $\underset{1}{\overset{8}{\int }}\left(\sqrt[3]{r}+\frac{1}{\sqrt[3]{r}}\right)\mathrm{dr}$
21. $\underset{-1}{\overset{0}{\int }}{\left(x+1\right)}^3\mathrm{dx}$
22. $\underset{-5}{\overset{-2}{\int }}\frac{x^4-1}{x^2+1}\mathrm{dx}$
23. $\underset{1}{\overset{e}{\int }}\frac{x^2+x+1}{x}\mathrm{dx}$
24. $\underset{4}{\overset{9}{\int }}{\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)}^2\mathrm{dx}$
25. $\underset{0}{\overset{1}{\int }}\left(\sqrt[4]{x^5}+\sqrt[5]{x^4}\right)\mathrm{dx}$
26. $\underset{1}{\overset{8}{\int }}\frac{x-1}{\sqrt[3]{x^2}}\mathrm{dx}$

Problem C. Definite integrals with trigonometric functions

1. $\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{3}}{\int }}\sin t\mathrm{dt}$
2. $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\cos \theta +2\sin \theta \right)d\theta$
3. $\underset{\frac{\pi }{2}}{\overset{\pi }{\int }}\sec x\tan x\mathrm{dx}$
4. $\underset{\frac{\pi }{3}}{\overset{\frac{\pi }{2}}{\int }}\csc x\cot x\mathrm{dx}$
5. $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}{\csc }^2\theta d\theta$
6. $\underset{\frac{\pi }{4}}{\overset{3}{\int }}{\sec }^2\theta d\theta$
7. $\underset{1}{\overset{\sqrt{3}}{\int }}\frac{6}{1+x^2}\mathrm{dx}$
8. $\underset{0}{\overset{0.5}{\int }}\frac{\mathrm{dx}}{\sqrt{1-x^2}}$

Problem D. Definite integrals with other functions

1. $\underset{4}{\overset{8}{\int }}\frac{1}{x}\mathrm{dx}$
2. $\underset{\ln 3}{\overset{\ln 6}{\int }}8e^x\mathrm{dx}$
3. $\underset{8}{\overset{9}{\int }}{2}^t\mathrm{dt}$
4. $\underset{-e^2}{\overset{-e}{\int }}\frac{3}{x}\mathrm{dx}$
5. $\underset{-2}{\overset{3}{\int }}\left|x^2-1\right|\mathrm{dx}$
6. $\underset{-1}{\overset{2}{\int }}\left(x-2\left|x\right|\right)\mathrm{dx}$
8. $\underset{0}{\overset{2}{\int }}\left(x^2-\left|x-1\right|\right)\mathrm{dx}$
9. $\underset{0}{\overset{2}{\int }}f\left(x\right)\mathrm{dx}$ where $f\left(x\right)=\left\{\begin{array}{rc}{x}^4& \text{ if }0\le x<1,\\ x^5& \text{ if }1\le x\le 2.\end{array}\right.$
10. $\underset{-\pi }{\overset{\pi }{\int }}f\left(x\right)\mathrm{dx}$ where $f\left(x\right)=\left\{\begin{array}{rc}x& \text{ if }-\pi \le x\le 0,\\ \sin x& \text{ if }0<x\le \pi .\end{array}\right.$

Problem F. Finding areas bounded by lines and a curve

1. Finding the area of the region bounded by the curve $xy-3x-2y-10=0,$ the $x$-axis, and the lines $x=3$ and $x=4.$
2. Find the area lying below the $x$-axis and above the parabola $y=4x+x^2.$
3. Graph the curve $y=2\sqrt{9-x^2}$ and determine the area enclosed between the curve and the $x$-axis.
4. Find the area bounded by the curve $y=x\left(x-3\right)\left(x-5\right),$ the $x$-axis and the lines $x=0$ and $x=5.$
5. Find the area enclosed between the curve $y=\sin 2x,0\le x\le \frac{\pi }{4}$ and the $x$-axes.
6. Find the area enclosed between the curve $y=\cos 2x,0\le x\le \frac{\pi }{4}$ and the axes.
7. Find the area enclosed between the curve $y=3\cos x,0\le x\le \frac{\pi }{2}$ and the axes.
8. Show that the ratio of the areas under the curves $y=\sin x$ and $y=\sin 2x$ between the lines $x=0$ and $x=\frac{\pi }{3}$ is 2 3 .
9. Find the area enclosed between the curve $y=\cos 3x,0\le x\le \frac{\pi }{6}$ and the axes.
10. Find the area enclosed between the curve $y={\tan }^{2}x,0\le x\le \frac{\pi }{4}$ and the axes.
11. Find the area enclosed between the curve $y={\csc }^{2}x,0\le x\le \frac{\pi }{4}$ and the axes.
12. Compare the areas under the curves $y={\cos }^{2}x$ and $y={\sin }^{2}x$ between $x=0$ and $x=\pi .$
13. Graph the curve $y=\frac{x}{\pi }+2{\sin }^{2}x$ and find the area between the $x$-axis, the curve and the lines $x=0$ and $x=\pi .$
14. Find the area bounded by $y=\sin x$ and the $x$-axis between $x=0$ and $x=2\pi .$

Problem G. Areas between curves

1. Find the area of the region bounded by the parabola $y^2=4x$ and the line $y=2x.$
2. Find the area bounded by the curve $y=x\left(2-x\right)$ and the line $x=4y-2.$
4. Calculate the area of the region bounded by the parabolas $y=x^2$ and $x=y^2.$
5. Find the area of the region bounded by the curves $y=\sqrt{x}$ and $y=x.$
6. Find the area of the region included between the parabola $y^2=x$ and the line $x+y=2.$
7. Find the area of the part of the first quadrant which is between the parabola $y^2=3x$ and the circle $x^2+y^2-6x=0.$
8. Find the area of the region between the curves $y^2=4x$ and $x=3.$
9. Use intergration to find the area of the triangular region bounded by the lines $y=2x+1,y=3x+1$ and $x=4.$
10. Find the area bounded by the parabola $x^2-y=2$ and the line $x+y=0.$
11. Find the area bounded by the curves $y=3x-x^2$ and $y=x^2-x.$
12. Graph the curve $y=\frac{1}{2}{x}^2+1$ and the straight line $y=x+1$ and find the area between the curve and the straight line.
13. Find the area of the region between the parabolas $4y^2=9x$ and $3x^2=16y.$
14. Find the area of the region between the curves $x^2+y^2=2$ and $3x=y^2.$
15. Find the area of the region between the curves $x^2+4\left(y-1\right)=0$ and $3y=x^2.$
16. Find the area of the region between the circles $x^2+y^2=4$ and ${\left(x-2\right)}^2+y^2=4.$
17. Find the area of the region enclosed by the parabola $y^2=4ax$ and the line $y=mx.$
18. Find the area between the parabolas $y=4ax$ and $y^2=4ax$
19. Find the area of the region between the circles $x^2+y^2=1$ and ${\left(x-1\right)}^2+y^2=1.$
20. Find the area bounded by the curves $y=x$ and $y=x^3.$
21. Graph $y=\sin x$ and $y=\cos x$ for $0\le x\le \frac{\pi }{2}$ and find the area enclosed by them and the $x$-axis.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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