Math 152 “FREE” Final Exam Review
Compiled by Ben Aurispa and Amy Austin
1. Integrate the following.
Z
x
√
(a)
dx
x−9
Z
8x − 7
(b)
dx
(x − 1)(x2 + 4)
Z
1
√
dx
(c)
2
x x2 − 4
Z
(d)
sec7 θ tan3 θ dθ
(e)
Z
sin2 (ln x)
dx
x
Z
x7 cos(x4 ) dx
Z ∞
1
3
dx
−
(g)
3x + 4 x + 7
0
Z 1
1
(h)
dx
4x − 1
0
Z ∞
5 + e−2x
dx diverges.
2. Show
x
1
(f)
3. Set up integrals to find the following.
(a) The area bounded by the curves y =
√
x, y = x2 , x = 0, and x = 4.
(b) The area bounded by the curves x = 2y 2 , x + y = 1.
π
(c) The volume of the solid found by rotating the region bounded by y = tan x, y = 0, x = 0 and x = about
4
x-axis.
√
(d) The volume of the solid found by rotating the region bounded by y = x, x = 0 and y = 4 about the
y-axis.
(e) The volume of the solid found by rotating the region bounded by y = 4 − x2 and y = 3 about the line x = 4.
(f) The volume of the solid found by rotating the region bounded by y = x2 and y = 2x + 3 about the line
y = −2.
(g) The volume of the solid whose base is the triangular region with vertices (0, 0), (3, 0), and (0, 4) and whose
cross sections perpendicular to the y-axis are semicircles.
(h) The length of the curve x = y 3 − y from y = 2 to y = 3.
(i) The surface area of the solid found by rotating the curve y =
√
x, 0 ≤ x ≤ 9 about the x-axis.
(j) The surface area of the solid found by rotating the curve y = t3 , x = t2 , 1 ≤ t ≤ 3 about the y-axis.
4. Find the average value of the function f (x) = cos2 x on the interval [0, π4 ].
5. A spring has a natural length of 3 ft. A force of 30 lbs is required to hold the spring stretched to a length of 5
ft. How much work is required to stretch the spring from 4 ft to 8 ft?
6. A trough is 20 m long and has semicicular ends with height 9 m. The trough has a 2 m spout at the top and is
full of water. Set up an integral to find the work necessary to pump all the water out of the spout.
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7. Determine whether the following series converge or diverge. Justify your answer. If the series converges, does it
also converge absolutely?
∞
X
(a)
n2
√
n7 + 12
n=1
(b)
∞
X
(2n)!
(n!)2
n=0
(c)
∞
X
(−1)n
√
3
n2 + 5
n=0
(d)
∞
X
(−1)n+1 n
ln n
n=2
(e)
∞
X
ln n
n2
n=3
8. Find the sums of the following series, if possible.
∞ X
2
2
−
(a)
n+1 n
n=3
(b)
∞
X
3(−2)n+1
5n
n=2
(c)
∞
X
(−1)n+1 42n+1
32n (2n)!
n=0
9. Find the radius and interval of convergence for the series
∞
X
32n (3x − 1)n
√
n
n=3
10. Express the following functions as Maclaurin series.
7x
5 − x3
(b) x sin(3x5 )
(a)
(c) x4 ln(1 + 4x2 )
1
(d)
(1 − 3x)2
11. Integrate the following as a power series.
Z
6
x4 e−x dx
12. Find the Taylor series for the function f (x) =
1
centered at x = 1.
(7x + 1)2
13. Consider the triangle formed by the vertices P (−1, −5, 2), Q(−4, −2, 1), and R(−1, 0, 2).
(a) Find the angle at vertex Q.
(b) Find the area of this triangle.
(c) Find a unit vector perpendicular to this triangle.
−−→
−→
(d) Find the vector and scalar projection of P Q onto P R
14. Find the volume of the parallelipiped determined by the vectors h1, 0, 2i, h0, 2, 3i, and h8, −5, 6i.
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