Math 220 (Fall 2009) Final Exam Name:

Instructions: Fully explain your answers and show your work. If you are not sure if you havewritten enough, please ask.

1. (12 pts.) Set up (but do not evaluate) integrals that give the following quantities:

(a) The volume of the solid of revolution obtained by revolving the region enclosed by thex-axis and the graph y = 2x− x2 about the line x = −1.

(b) The work required to pump half of the water out over the top of a full swimming poolwhich is 8 feet wide, 6 feet deep, and 20 feet. (Assume the density of water is 62.5pounds per cubic foot.)

(c) The length of the part of the graph y = 13x3 for 0 ≤ x ≤ 1.

2. (6 pts.) Let R be the region between the graph y =√x and the x-axis, for 0 ≤ x ≤ 4. Find

the coordinates of the center of mass of this region.

3. (5 pts.) Solve the initial value problemdy

dx=

6− 2y

x2 − 1, y(0) = 5.

4. (a) (5 pts.) Find the approximate value of∫ .8

0ln(1+x2) dx using Simpson’s rule, with n = 4

intervals.

(b) (5 pts.) Find the approximate value of this integral, with error less than 1/100, usinga power series.

5. (16 pts.) Find the following antiderivatives:

(a)

∫x2 coshx dx

(b)

∫sec3 x tanx dx

(c)

∫x− 3

x3 + 9xdx

(d)

∫1√

x2 + 4dx

6. (8 pts.) Evaluate the improper integral or show that it diverges:

(a)

∫ 2

1

1

x− 1dx

(b)

∫ ∞

2

1

x(lnx)2dx

7. (10 pts.) Explain why each of the following series converge, find the approximate sum ofthe series using the first 4 terms, and estimate the remainder.

(a)∞∑

k=1

1

k 3k

(b)∞∑

n=1

√2n− 1

n2

8. (10 pts.) Determine if the following series converge absolutely, converge conditionally, ordiverge.

(a)∞∑

n=1

(−1)n√n2 + 1

(b)∞∑

n=1

(−1)n(

n

2n− 1

)n

9. (a) (5 pts.) Calculate the Taylor polynomial T3(x) of degree 3, centered at 2, for f(x) =1/x2.

(b) (6 pts.) Use Taylor’s Inequality to find an upper bound for the remainder for T3(x)when 1 ≤ x ≤ 3.

10. (6 pts.) Determine the radius and interval of convergence for the power series∑∞

n=1

2n(x+ 1)n

n2

11. (6 pts.) Find the Maclaurin series for f(x) = arcsin(x); express your answer in summationnotation, and find the first 4 nonzero terms.

12. (10 pts.) For the parametrized curve defined by x = t2 + t, y = t3 − t, find

(a) the slope of the tangent when the curve passes through the origin;

(b) the values of t for which the curve is concave up;

(c) an integral that computes the length of the loop formed by the curve. (Do not evaluatethis integral.)

13. (10 pts.) Sketch the curve defined by the polar equation r = 1 + 2 sin θ, for 0 ≤ θ ≤ 2π onthe given grid, and find the area enclosed by the curve inside the first quadrant.

1 2 3

Π

4

Π

3

Π

6

Name Statement

Divergence Test If limk→∞ ak 6= 0, then∞∑

k=1

ak diverges. (If limk→∞ ak = 0, there is no conclusion.)

Integral Test

Let∞∑

k=1

ak be a series with positive terms, such that ak = f(k) for a continuous

function f(x) which is decreasing for x sufficiently large. Then∑ak and

∞∫1

f(x) dx

either both converge or both diverge.

Comparison TestLet

∑ak and

∑bk be series with non-negative terms such that ak ≤ bk. If

∑bk

converges, then∑ak converges. If

∑ak diverges, then

∑bk diverges.

Limit Comparison TestLet

∞∑

k=1

ak and

∞∑

k=1

bk be two series with positive terms, and let L = limk→∞ ak/bk.

If L is positive and finite, then either both series converge or both diverge.

AlternatingSeries Test

If∞∑

k=1

uk is an alternating series with |uk| decreasing and limk→∞ uk = 0, then the

series converges.

Ratio Test& Root Test

Let∑uk be any series, and let L = limk→∞ |uk+1|/|uk|, or let L = limk→∞ k

√|uk|.

If L < 1 the series is absolutely convergent; if L > 1 the series diverges; if L = 1there is no conclusion.

sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ

1 + cot2 θ = csc2 θ sin 2x = 2 sinx cosx

cos2 x = 12 (1 + cos 2x) sin2 x = 1

2 (1− cos 2x)

sinhx = 12 (ex − e−x) coshx = 1

2 (ex + e−x)

(sinx)′ = cosx (cosx)′ = − sinx

(tanx)′ = sec2 x (secx)′ = secx tanx

(cscx)′ = − cscx cotx (cotx)′ = − csc2 x

(arcsinx)′ =1√

1− x2(arctanx)′ =

1

1 + x2∫tanx dx = ln | secx|+ C

∫secx dx = ln | secx+ tanx|+ C

∫secn x dx =

1

n− 1secn−2 x tanx+

n− 2

n− 1

∫secn−2 x dx

∫u dv = uv −

∫v du

1

1− x =∞∑

n=0

xn for − 1 < x < 1

ex =∞∑

n=0

xn

n!= for all x (1 + x)k =

∞∑

n=0

(kn

)xn for − 1 < x < 1

sinx =

∞∑

n=0

(−1)nx2n+1

(2n+ 1)!= for all x cosx =

∞∑

n=0

(−1)nx2n

(2n)!= for all x

arctan(x) =∞∑

n=0

(−1)nx2n+1

2n+ 1for − 1 < x < 1 ln(1 + x) =

∞∑

n=1

(−1)n−1xn

n= for − 1 < x < 1