MAT 152, Calculus II
Final Exam Preparation Sheet
Problem 1. Find f(x) if
f00(x) = 1
2cosh(x/2) + 4
(2x+ 1)2, f0(0) = 0 ,and f(0) = 0 .
Problem 2. Find f0(x)
(a)f(x) = (cosh(5x))sin(4x)(b)f(x) = Zarctan(3x)
2 ln(x)qet+ tan2t dt
Problem 3. Evaluate the integral Rsin(5x)s in(2x)dx
Problem 4. (i) Sketch the region bounded by the curves y=xe−x/3,y= 0 and x= 0; x≥0.
(ii) Find the volume generated by rotating the region about the y-axis.
Problem 5. Find the volume of the resulting solid if the region under the curve y=10
(x+3)(x2+1)
from x= 0 to x=3
4is rotated about the y-axis.
Problem 6. Find the length of the curve
y= 3 + x2
4−ln 2x
2,1≤x≤√e
Problem 7. Find the area of the surface obtained by rotating the curve x= 4+ y2
6,0≤y≤4
about the x-axis.
Problem 8. Find the exact length of the polar curve
r=e2θ,0≤θ≤ln 3 .
Problem 9. Determine whether the series is convergent or divergent. If it is convergent, find
its sum P∞
n=0 2nen+1
6n−1
Problem 10. Use the Ratio Test to determine whether the series is absolutely convergent,
conditionally convergent, or divergent. P∞
n=1 (−1)nn!5n
2n+1nn
Problem 11. Express the function as the sum of a power series by first using partial fractions.
Find the interval of convergence f(x) = 7
2x2−5x−3
Problem 12. Evaluate the indefinite integral as an infinite series R3
√2x3+ 3 dx
Bonus Problem 1. Find the volume of the resulting solid if the region under the curve y=
10
(x+3)(x2+1) from x= 0 to x=∞is rotated about the y-axis.
Bonus Problem 2. The Bessel function of order 0 is defined by
J0(x) =
∞
X
n=0
(−1)nx2n
n!(n)!22n
Show that J0satisfies the differential equation
x2J00
0(x) + xJ0
0(x) + x2J0(x) = 0
1