Math 152/172
WEEK in REVIEW 1.
1. Evaluate the integral
Z π/12
a)
sin(3x − 2) dx
b)
Z
x(4x2 + 1)6 dx
d)
Z
√
sin x
√
dx
x
f)
Z
x + arcsin x
√
dx
1 − x2
h)
Z
ex
dx
ex + 1
0
c)
Z
e)
Z 1
3
2
3
x (x + 3) dx
3
x2 ex dx
0
2x2 + 4x
dx
x3 + 3x2 − 4
g)
Z
i)
Z e4
dx
√
x ln x
j)
Z
x dx
√
1 + x4
Z 4
1
x2
r
l)
Z
tan x ln(cos x) dx
e
k)
1
1
+ 1 dx
x
Spring 2025.
2. Sketch the region bounded by the given curves and find the area of the region.
(a) y = x2 + 2, y = 2x + 5, x = 0, x = 6.
(b) y = cos x, y = sin 2x, x = 0, xπ/2.
(c) x + y 2 = 2, x + y = 0.
3. Find the area of the triangle with vertices (0, 0), (3, 1), (6, −6).
4. Find the area of the region bounded by the parabola y = x2 , the tangent line to this parabola at (1,1),
and the x-axis.
5. Find the volume of the solid S whose base is a parabolic region {(x, y)|x2 ≤ y ≤ 1}, and cross sections
perpendicular to the y-axis are equilateral triangles.
6. Find the volume of the solid S whose base is the triangular region with vertices (0,0), (2,0), (0,1), and
cross sections perpendicular to the x-axis are semicircles.
7. Find the volume of a frustum of a pyramid with square base of side b, square top of side a, and height h.
1