User:Melikamp/calc

Review of Anti-derivatives

1. Find $\displaystyle \int \left(2x^{2}-{\frac {1}{x}}\right)dx$

2. Find $\displaystyle \int \left(x^{-1.5}-x^{6/7}\right)dx$

3. Find $\displaystyle \int \sin(3x+5)dx$

4. Find $\displaystyle \int {\frac {dx}{2x+1}}$

5. Find $\displaystyle \int {\frac {2}{1+25x^{2}}}dx$

6. Find $\displaystyle \int 3x^{2}(x^{3}-5)^{8}dx$

Integration by Parts

10. Find $\displaystyle \int x^{2}e^{x}dx$

11. Find $\displaystyle \int {\frac {\ln(x)}{x^{3}}}dx$

12. Find $\displaystyle \int e^{x}\cos(x)dx$

Integration by Partial Fractions

13. Find $\displaystyle \int {\frac {9x+45}{x^{2}+5x-14}}dx$

14. Find $\displaystyle \int {\frac {x^{4}-5x^{3}-x^{2}+4x+17}{x^{2}-4x-5}}dx$

15. Find $\displaystyle \int {\frac {5x^{2}-2x-18}{(2x+1)(x-1)(x+4)}}dx$

Improper Integrals

20. Find $\displaystyle \int _{-1/2}^{4}{\frac {1}{\sqrt {2x+1}}}dx$

21. Find $\displaystyle \int _{2}^{\infty }xe^{-x}dx$

22. Find $\displaystyle \int _{0}^{\infty }\sin(x)\cos(x)dx$

23. Find $\displaystyle \int _{0}^{\infty }{\frac {1}{x^{2}+9}}dx$

24. Find $\displaystyle \int _{0}^{e^{2}}\ln(x)dx$

Integration Review

30. Find $\displaystyle \int 4x\sin(1-x^{2})dx$

31. Find $\displaystyle \int _{2}^{\infty }e^{1-x}dx$

32. Find $\displaystyle \int {\frac {3x^{3}-x^{2}-8x-1}{x^{2}-x-2}}dx$

33. Find $\displaystyle \int {\frac {4x\ln(x^{2}+1)}{x^{2}+1}}dx$

34. Find $\displaystyle \int _{0}^{\infty }{\frac {2}{(x+3)(x+5)}}dx$

Distance Traveled and Arc Length in Space

40. Find the distance traveled by the particle with position function $\displaystyle \mathbf {p} (t)=(2\sin(t),5t,2\cos(t))$ for $t\in [-10,10]$.

41. Find the distance traveled by the particle with position function $\displaystyle \mathbf {p} (t)=(12t,8t^{3/2},3t^{2})$ for $t\in [0,1]$.

42. Find the distance traveled by the particle with position function $\displaystyle \mathbf {p} (t)=(t^{2},-2t,\ln(t))$ for $t\in [1,e]$.

43. Find the arc length of the graph of the function $\displaystyle y(x)=ax+b$, where $a\neq 0$, for $x\in [c,d]$.

44. Find the arc length of the graph of the function $\displaystyle y(x)=x^{3/2}$ for $x\in [0,5/9].$

Area Swept Out

50. Find the area swept out by a particle moving along the parametrized curve $x=t$, $y=t^{3}$, for $t\in [-1,1]$. Plot the curve and shade the area swept out before setting up the integral.

51. Find the area swept out by a particle moving along the parametrized curve $x=t^{3}$, $y=t^{2}$, for $t\in [-2,2]$. Plot the curve and shade the area swept out before setting up the integral.

52. Plot the polar curve $r=\sin(2\theta )$ for $\theta \in [0,2\pi ]$, and find the area enclosed by it.

53. Plot the polar curve $r=1+\cos(\theta )$ for $\theta \in [0,2\pi ]$, and find the area enclosed by it.

54. Plot the polar curve $r=\cos(5\theta )$ for $\theta \in [0,\pi ]$, and find the area enclosed by it.

Volume

60. Let $S$ be the region in the first quadrant above the $x$-axis and below the curve $y=x^{n}$, $x\in [0,1]$, $n$ a positive ingeter. Find the volume of the solid obtained by revolving $S$ about the $x$-axis.

61. Let $S$ be the region above the line $y=x-2$ and below the line $y=2-x$, $x\in [0,2]$. Find the volume of the solid obtained by revolving $S$ about the $y$-axis.

Mass and Density

70. Find the mass of a stick extending along the $x$-axis from $x=1$ to $x=9$ if the linear density of the stick is $\delta _{l}(x)=1+x^{-1}$ (assume SI units). Write an equation for the mass midpoint.

71. Find the mass of the thin plate lying in the $xy$-plane below the curve $y=1-x^{2}$ and above the curve $y=x^{2}-1$ if the area density is $\delta _{ar}(x)=x+2$.

Center of Mass and Moments

80. Find the center of mass of a thin wire extending from $x=1$ to $x=e$ along the x-axis if the linear density of the wire is $\delta _{l}(x)=ln(x)$.

81. Find the center of mass of a thin plate occupying the region $S$ in the xy-plane, if $S$ is a region below the curve $y=x^{2}$ and above the curve $y=x^{3}$, with $x\in [0,1]$, and the area density of the plate is $\delta _{ar}(x)=(x+1)$.

Work and Energy

90. Suppose that we have a tank, which is a right circular cylinder of radius 1 meter and height 4 meters, and the tank is initially filled half-way. Find the amount of work required to pump all of the water to the top of the tank. Use $1000\ \mathrm {kg} /\mathrm {m} ^{3}$ as the density of water and $g=9.8\ \mathrm {m} /\mathrm {s} ^{2}$ as the gravity of Earth.

91. Suppose that a bucket is lifted to the top of a building 12 meters high at a constant rate of $1\ \mathrm {m} /\mathrm {s}$. The initial weight of the bucket is $14\ \mathrm {kg}$, and it is leaking sand at the rate of $0.2\ \mathrm {kg} /\mathrm {s}$. Find the work required to lift the bucket. Use $g=9.8\ \mathrm {m} /\mathrm {s} ^{2}$ as the gravity of Earth.

Taylor Series

100. Find the Taylor polynomial of 8th degree, centered at zero, for the function $f(x)=\cos(x)$, and make a guess about the corresponding Taylor series.

101. Find the Taylor polynomial of 8th degree, centered at zero, for the function $x\sin(x)$, and make a guess about the corresponding Taylor series.

102. Find the Taylor series for the function $f(x)=e^{x}$ centered at $a=1$.

103. Find the Taylor polynomial of 3rd degree for the function $f(x)={\sqrt {x}}$, centered at $a=1$.

Cross Product

110. Find the area of the triangle with vertices $(1,2,3)$, $(-1,-1,2)$, and $(3,0,-3)$.

111. Find a standard equation for the plane containing both the point $(5,0,2)$ and the line $(x,y,z)=(3,0,1)+t(2,1,2)$.

Multi-Component Functions of a Single Variable

120. Let the position of the particle be given by $\mathbf {p} (t)=(2\sin(t),2\cos(t))$. Find velocity, acceleration, and speed of the particle as functions of $t$. Sketch the path of the particle, and draw the velocity vector $\mathbf {v} (\pi /4)$ and the acceleration vector $\mathbf {a} (\pi /4)$.

121. Find the distance traveled by a particle between times $t_{0}=0$ and $t_{1}=1$ if the position function of the particle is $\mathbf {p} (t)=(e^{t},{\sqrt {2}}t,e^{-t})$.

$e-e^{-1}$

Directional Derivative

130. Find the total derivative $d_{\mathbf {p} }f(\mathbf {v} )$ of $f(x,y,z)=xz^{2}+\sin(y)$ at the point $\mathbf {p} =(-1,0,2)$, if $\mathbf {v} =(2,-1,-3)$.

131. Find the value of the directional derivative of $f(x,y,z,w)=xy^{2}-zw$ at $\mathbf {p} =(1,2,3,4)$ in the direction of the vector $(1,3,5,1)$.

132. Let $f(x,y)=x^{2}-3y^{2}$ and $\mathbf {p} =(5,-2)$.

1. Find the direction $\mathbf {u}$ in which the directional derivative of $f$ at $\mathbf {p}$ is maximized, and the value of $D_{\mathbf {u} }f(\mathbf {p} )$.
2. Find the directions for which the directional derivative of $f$ at $\mathbf {p}$ is zero.