Antiderivatives
Integration
Applications
Sequences and Series
Series
100
Consider the graph of the function f(x) shown below. 
a) Estimate the integral
∫07f(x)dx≈
b) If F is an antiderivative of the same function f and F(0)=40, estimate F(7):
F(7)≈
a) -19
b) 21
100
For the following integral, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integral.
∫xsinxdx
A. integration by parts
B. neither
C. substitution
A. integration by parts
100
Find the area of the region between y=x1/2 and y=x1/4 for 0≤x≤1.
Area =∫01(x1/4−x1/2)dx=0.133
100
Find a formula for sn, n≥1 for the sequence -3, 5, -7, 9, -11...
sn=
sn=(−1)n(2n+1)
100
Carefully determine the convergence of the series ∑∞n=1(−1)n/3n. The series is
A. absolutely convergent
B. conditionally convergent
C. divergent
A. absolutely convergent
200
Find the derivative: d/dx∫xacos(tan(t))dt=
−cos(tan(x))
200
4. ∫x2cos(x3)dx
A. substitution
B. neither
C. integration by parts
A. substitution
200
Find the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x6, y=1, and the y-axis around the y-axis.
Volume =
3pi/4
200
For the sequence below, enter either diverges if the sequence diverges, or the limit of the sequence if the sequence converges as n→∞
4n
diverges
200
Carefully determine the convergence of the series ∑∞n=1(−1)n/3n. The series is
A. absolutely convergent
B. conditionally convergent
C. divergent
B. conditionally convergent
300
Find the following integral. Note that you can check your answer by differentiation.
∫t3(t4−3)3dt=
1/16(t4−3)4+C
300
∫1/((x+6)(x+8))dx=
hint: (A/x+6)+(B/x+8)
0.5ln(|x+6|)−0.5ln(|x+8|)+C
300
A rod has length 3 meters. At a distance x meters from its left end, the density of the rod is given by δ(x)=4+5x g/m.
(a) Complete the Riemann sum for the total mass of the rod (use Dx in place of Δx):
mass = Σ
(b) Convert the Riemann sum to an integral and find the exact mass.
mass =
(a) Σ(4+5x)Δx
(b) 69/2 grams
300
For the sequence below, enter either diverges if the sequence diverges, or the limit of the sequence if the sequence converges as n→∞
4n+8/n2
converges to 0
300
Find the Taylor polynomials of degree n approximating cos(3x) for x near 0:
For n=2, P2(x)=
1−32/2x2
400
Find the general antiderivative F(x) of the function f(x) given below. Note that you can check your answer by differentiation.
f(x)=2x3sin(x4)
antiderivative F(x)=
−1/2cos(x4)+C
400
Using a fixed number of subdivisions, we approximate the integrals of f and g on the interval shown in the figure below. 
(The function f(x) is shown in blue, and g(x) in black; click on the graph to get a larger version.)
For which function, f or g is LEFT more accurate?
A. f
B. g
A. f
400
A fuel oil tank is an upright cylinder, buried so that its circular top is 10 feet beneath ground level. The tank has a radius of 7 feet and is 21 feet high, although the current oil level is only 17 feet deep. Calculate the work required to pump all of the oil to the surface. Oil weighs 50lb/ft3
Work =
937125pi lbf
400
Find the sum of the series
1+13+19+...+1/3n-1+...
1.5
400
Find the Taylor polynomials of degree n approximating cos(3x) for x near 0:
For n=4, P4(x)=
1−32/2x2+34/4!x4
500
Find the following integral. Note that you can check your answer by differentiation.
∫ln7(z)/z dz=
1/8(ln(z))8+C
500
Using the figure showing f(x) below, order the following approximations to the integral ∫03f(x)dx and its exact value from smallest to largest. 
Enter each of "LEFT(n)", "RIGHT(n)", "TRAP(n)", "MID(n)" and "Exact" in one of the following answer blanks to indicate the correct ordering:
_____ < _____ < _____ < _____ < _____
LEFT(n)<MID(n)<Exact<TRAP(n)<RIGHT(n)
500
Consider the integral
∫03−8/(x√x)dx
If the integral is divergent, type an upper-case "D". Otherwise, evaluate the integral.
D
500
Let
an=2n/10n+7
For the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter 'infinity' if it diverges to ∞, '-infinity' if it diverges to −∞ or 'DNE' otherwise.
(a) The series ∑n=1∞2n/10n+7
(b) The sequence {2n/10n+7}
(a) ∞
(b) 1/5
500
Represent the function 4/(1−10x) as a power series f(x)=∑∞n=0cnxn
c0=
c1=
c2=
c3=
c4=
Find the radius of convergence R=
c0=4
c1=40
c2=400
c3=4000
c4=40000
R=0.1