MATH 152 Spring 2016
Final Exam Review - Part 1
1. Setup an integral to find the area bounded by 2x + y = 3, y = x2 = 4x,
x = −1, and x − 4.
√
2. Consider the region bounded by y = x2 and y = 8 x.
(a) Setup an integral to represent the volume of the solid obtained by
rotating the region around the line y = −1.
(b) Setup an integral to represent the volume of the solid obtained by
rotating the region around the line x = 5.
3. Setup an integral that represents the volume of a solid whose base is
bounded by y = 4 − x2 and the x-axis, and whose cross-sections perpendicular to the x-axis are semicircles.
4. Consider the region bounded by y = 2 + ln x, y = cos x, x = 1, and x = π.
Setup an integral gives the volume of the solid obtained by rotating this
region about the line
(a) x = π
(b) y = −4
5. An inverted conical shaped tank is full of water. If the cone has height 5
m and radius 2 m, setup an integral represents the amount of work done
in pumping all of the water out of a spout 2 m from the top of the cone.
Note: pg = 9800 Newtons per cubic meter gives the weight density of
water.
6. A spring has a natural length of 2 m. It requires 25 J of work to stretch
the spring from 2 m to 7 m. How much work is done in stretching the
spring from 3 m to 5 m?
7. Find the average value of the function f (x) = sin4 x cos x on the interval
[0, π6 ].
8. Evaluate the following integral:
Z π4
(a)
sin4 x cos3 x dx.
0
Z
(b) 3x2 e−2x dx
Z
sin x
(c)
dx
e4 cos x
Z
(d) tan3 x sec5 x dx.
Z
(e)
tan2 x dx.
(f)
R3
3
0 2x−4
∞
dx
Z
ln(x)
dx.
x3
1
Z
1
√
dx
(h)
x4 x2 − 9
Z
2
(i)
dx
(x − 1)(x2 + 1)
(g)
1
9. Consider the curve ey = x2 from 1 ≤ x ≤ e3 .
(a) Setup an integral in terms of x for the length of the curve.
(b) Setup an integral with variable y for the length of the curve.
10. Find the surface area of the curve y 6 = x3 from (0, 0) to (1, 1) rotated
around the x-axis.
11. Setup an integral to find the surface area obtained by rotating the curves
around the given axis:
(a) y = sec x2 , 0 ≤ x ≤ π4 , around the y-axis.
(b) y = e2t and x = 3e3t from 0 ≤ t ≤ 4, around the x-axis.
(c) y = ln(5x + 2) from 1 ≤ x ≤ 7, around the x-axis.
12. Find the unit vector in the direction of 2a − 3b for a = h−1, 4, 3i and
b = h2, 3, 1i.
13. For the points P (7, −4, 2), Q(3, 1, 0), and R(3, 4, 5), find the angle between
−−→
−→
the vectors P Q and P R.
14. Find the area of the triangle with vertices P (2, 3, −1), Q(3, 5, 7), and
R(0, 4, 9).
15. Find the scalar and vector projection of b = h2, −3, 4i onto a = h4, 1, −2i.
16. Find the cross product a × b for a = h3, 5, 4i and b = h0, −3, 2i.
17. Find the volume of the parellelepiped determined by the vectors u =
h3, −5, 1i, v = h3, 1, 1i, and w = h0, 2, −2i.
18. Given the sequence an = ne−n , n ≥ 1, which of the following statements
are true?
I. an is increasing.
II. (−1)n an converges
III. an is
bounded.
19. If the nth partial sum of the series
the sum, S, of the series
∞
X
∞
X
an is sn =
4n−2
n+7 ,
find a3 as well as
n=1
an .
n=1
20. For the series
∞
X
n=1
sin
nπ 6
find s3 .
21. Given the sequence a1 = 5 and an+1 =
bounded, what statement is true about an ?
2
√
4an + 3 is increasing and