Math 152 “FREE” Final Exam Review
Disclaimer: This review by no means covers every concept covered in Math 152 and should not be used as
your only source of study. It is a highlight of some of the material.
Solutions will be posted at www.math.tamu.edu/∼baurispa/math152/homepage.html
1. Integrate the following.
x
dx
x−9
Z
8x − 7
(b)
dx
(x − 1)(x2 + 4)
Z
1
√
(c)
dx
2
x x2 − 4
Z
(a)
√
Z
sec7 θ tan3 θ dθ
Z
sin2 (ln x)
dx
x
Z
x7 cos(x4 ) dx
(d)
(e)
(f)
Z ∞
(g)
0
3
1
−
3x + 4 x + 7
dx
2. Set up integrals to find the following.
(a) The area bounded by the curves y = x3 , y = −x3 + 2, x = 0, and x = 6.
(b) The volume of the solid found by rotating the region bounded by y = 4 − x2 and y = 3 about
the line x = 4.
(c) The volume of the solid found by rotating the region bounded by y = x2 and y = 2x + 3 about
the line y = −2.
(d) 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.
(e) The length of the curve x = y 3 − y from y = 2 to y = 3.
(f) The surface area of the solid found by rotating the curve y =
√
x, 0 ≤ x ≤ 9 about the x-axis.
(g) The surface area of the solid found by rotating the curve y = t3 , x = t2 , 1 ≤ t ≤ 3 about the
y-axis.
3. Find the average value of the function f (x) = tan x on the interval [0, π4 ].
4. 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?
5. 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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6. Determine whether the following series converge or diverge. Justify your answer. If the series converges,
does it also converge absolutely?
(a)
(b)
(c)
(d)
(e)
∞
X
√
n=1
∞
X
n2
n7 + 12
(2n)!
(n!)2
n=0
∞
X
(−1)n
√
3
n=0
∞
X
n2 + 5
(−1)n+1 n
ln n
n=2
∞
X
ln n
n=3
n2
7. Find the sums of the following series, if possible.
(a)
(b)
(c)
∞ X
n=3
∞
X
2
2
−
n+1 n
3(−2)n+1
5n
n=2
∞
X
(−1)n+1 42n+1
n=0
32n (2n)!
8. Find the radius and interval of convergence for the series
∞
X
32n (3x − 1)n
√
n=3
n
9. Express the following functions as Maclaurin series.
7x
5 − x3
(b) x sin(3x5 )
(a)
(c) x4 ln(1 + 4x2 )
Z
10. Integrate the following as a power series.
6
x4 e−x dx
11. Find the Taylor series for the function f (x) =
1
centered at x = 1.
(7x + 1)2
12. Consider the triangle formed by the vertices P (−1, −5, 2), Q(−4, −2, 1), and R(−1, 0, 2).
(a) Find tha angle at vertex Q.
(b) Find the area of this triangle.
(c) Find a unit vector perpendicular to this triangle.
13. Find the volume of the parallelipiped determined by the vectors < 1, 0, 2 >, < 0, 2, 3 >, and
< 8, −5, 6 >.
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