User:Melikamp/calc

< User:Melikamp

Review of Anti-derivatives

1. Find

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

{\frac {2x^{3}}{3}}-\ln |x|+C

{\frac {2x^{3}}{3}}-\ln |x|+C

2. Find

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

{\frac {-2}{\sqrt {x}}}-{\frac {7}{13}}x^{13/7}+C

{\frac {-2}{\sqrt {x}}}-{\frac {7}{13}}x^{13/7}+C

3. Find

\displaystyle \int \sin(3x+5)dx

-{\frac {1}{3}}\cos(3x+5)+C

-{\frac {1}{3}}\cos(3x+5)+C

4. Find

\displaystyle \int {\frac {dx}{2x+1}}

{\frac {1}{2}}\ln |2x+1|+C

{\frac {1}{2}}\ln |2x+1|+C

5. Find

\displaystyle \int {\frac {2}{1+25x^{2}}}dx

{\frac {2}{5}}\tan ^{-1}(5x)+C

{\frac {2}{5}}\tan ^{-1}(5x)+C

6. Find

\displaystyle \int 3x^{2}(x^{3}-5)^{8}dx

{\frac {(x^{3}-5)^{9}}{9}}+C

{\frac {(x^{3}-5)^{9}}{9}}+C

Integration by Parts

10. Find

\displaystyle \int x^{2}e^{x}dx

x^{2}e^{x}-2xe^{x}+2e^{x}+C

x^{2}e^{x}-2xe^{x}+2e^{x}+C

11. Find

\displaystyle \int {\frac {\ln(x)}{x^{3}}}dx

-{\frac {\ln(x)}{2x^{2}}}-{\frac {1}{4x^{2}}}+C

-{\frac {\ln(x)}{2x^{2}}}-{\frac {1}{4x^{2}}}+C

12. Find

\displaystyle \int e^{x}\cos(x)dx

{\frac {e^{x}}{2}}(\cos x+\sin x)+C

{\frac {e^{x}}{2}}(\cos x+\sin x)+C

Integration by Partial Fractions

13. Find

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

7\ln |x-2|+2\ln |x+7|+C

7\ln |x-2|+2\ln |x+7|+C

14. Find

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

\displaystyle {\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}+2\ln |x-5|-3\ln |x+1|+C

\displaystyle {\frac {x^{3}}{3}}-{\frac {x^{2}}{2}}+2\ln |x-5|-3\ln |x+1|+C

15. Find

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

\displaystyle {\frac {3}{2}}\ln |2x+1|-\ln |x-1|+2\ln |x+4|+C

\displaystyle {\frac {3}{2}}\ln |2x+1|-\ln |x-1|+2\ln |x+4|+C

Improper Integrals

20. Find

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

3

3

21. Find

\displaystyle \int _{2}^{\infty }xe^{-x}dx

3e^{-2}

3e^{-2}

22. Find

\displaystyle \int _{0}^{\infty }\sin(x)\cos(x)dx

Diverges

Diverges

23. Find

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

{\frac {\pi }{6}}

{\frac {\pi }{6}}

24. Find

\displaystyle \int _{0}^{e^{2}}\ln(x)dx

e^{2}

e^{2}

Integration Review

30. Find

\displaystyle \int 4x\sin(1-x^{2})dx

2\cos(1-x^{2})+C

2\cos(1-x^{2})+C

31. Find

\displaystyle \int _{2}^{\infty }e^{1-x}dx

1/e

1/e

32. Find

\displaystyle \int {\frac {3x^{3}-x^{2}-8x-1}{x^{2}-x-2}}dx

3x^{2}/2+2x+\ln |x-2|-\ln |x+1|+C

3x^{2}/2+2x+\ln |x-2|-\ln |x+1|+C

33. Find

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

\displaystyle \left[\ln(x^{2}+1)\right]^{2}+C

\displaystyle \left[\ln(x^{2}+1)\right]^{2}+C

34. Find

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

\ln(5/3)

\ln(5/3)

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]

.

20{\sqrt {29}}

20{\sqrt {29}}

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]

.

15

15

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]

.

e^{2}

e^{2}

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]

.

(d-c){\sqrt {1+a^{2}}}

(d-c){\sqrt {1+a^{2}}}

44. Find the arc length of the graph of the function

\displaystyle y(x)=x^{3/2}

for

x\in [0,5/9].

19/27

19/27

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.

{\frac {1}{2}}

{\frac {1}{2}}

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.

{\frac {32}{5}}

{\frac {32}{5}}

52. Plot the polar curve

r=\sin(2\theta )

for

\theta \in [0,2\pi ]

, and find the area enclosed by it.

{\frac {\pi }{2}}

{\frac {\pi }{2}}

53. Plot the polar curve

r=1+\cos(\theta )

for

\theta \in [0,2\pi ]

, and find the area enclosed by it.

{\frac {3}{2}}\pi

{\frac {3}{2}}\pi

54. Plot the polar curve

r=\cos(5\theta )

for

\theta \in [0,\pi ]

, and find the area enclosed by it.

{\frac {\pi }{4}}

{\frac {\pi }{4}}

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.

\displaystyle {\frac {\pi }{2n+1}}

\displaystyle {\frac {\pi }{2n+1}}

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.

\displaystyle {\frac {16\pi }{3}}

\displaystyle {\frac {16\pi }{3}}

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.

Mass is

8+\ln 9

kg, mass midpoint is at

x_{m}

, where

x_{m}+\ln |x_{m}|=5+\ln 3

Mass is

8+\ln 9

kg, mass midpoint is at

x_{m}

, where

x_{m}+\ln |x_{m}|=5+\ln 3

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

.

{\frac {16}{3}}

kg

{\frac {16}{3}}

kg

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)

.

{\frac {e^{2}+1}{4}}

{\frac {e^{2}+1}{4}}

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)

.

\left({\frac {5}{8}},{\frac {83}{224}}\right)

\left({\frac {5}{8}},{\frac {83}{224}}\right)

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.

1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}

1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}

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.

x^{2}-{\frac {x^{4}}{3!}}+{\frac {x^{6}}{5!}}-{\frac {x^{8}}{7!}}

x^{2}-{\frac {x^{4}}{3!}}+{\frac {x^{6}}{5!}}-{\frac {x^{8}}{7!}}

102. Find the Taylor series for the function

f(x)=e^{x}

centered at

a=1

.

\sum _{k=0}^{\infty }{\frac {e(x-1)^{k}}{k!}}

\sum _{k=0}^{\infty }{\frac {e(x-1)^{k}}{k!}}

103. Find the Taylor polynomial of 3rd degree for the function

f(x)={\sqrt {x}}

, centered at

a=1

.

1+{\frac {(x-1)}{2}}-{\frac {(x-1)^{2}}{8}}+{\frac {(x-1)^{3}}{16}}

1+{\frac {(x-1)}{2}}-{\frac {(x-1)^{2}}{8}}+{\frac {(x-1)^{3}}{16}}

Cross Product

110. Find the area of the triangle with vertices

(1,2,3)

,

(-1,-1,2)

, and

(3,0,-3)

.

{\sqrt {138}}

{\sqrt {138}}

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)

.

x+2y-2z-1=0

x+2y-2z-1=0

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)

.

\mathbf {v} (t)=(2\cos(t),-2\sin(t))

,

|\mathbf {v} (t)|=2

,

\mathbf {a} (t)=(-2\sin(t),-2\cos(t))

\mathbf {v} (t)=(2\cos(t),-2\sin(t))

,

|\mathbf {v} (t)|=2

,

\mathbf {a} (t)=(-2\sin(t),-2\cos(t))

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}

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)

.

19

19

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)

.

-7/6

-7/6

132. Let

f(x,y)=x^{2}-3y^{2}

and

\mathbf {p} =(5,-2)

.

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

(a)

\displaystyle {\frac {(10,12)}{2{\sqrt {61}}}}

and

2{\sqrt {61}}

, (b)

\displaystyle \pm {\frac {(12,-10)}{2{\sqrt {61}}}}

(a)

\displaystyle {\frac {(10,12)}{2{\sqrt {61}}}}

and

2{\sqrt {61}}

, (b)

\displaystyle \pm {\frac {(12,-10)}{2{\sqrt {61}}}}

Retrieved from "https://en.wikibooks.org/w/index.php?title=User:Melikamp/calc&oldid=2626108"