Math 0230
Fall Semester 2022
Sample Exam 1
Instructions: No books, notes, calculators, internet, person, or any
other form of assistance may be used on this exam. Be sure to clearly justify
all of your answers for full credit. If no justification is provided, zero credit
will be awarded.
This sample exam is only a summary of problems that covers the material
and that you should be able to solve within 50minutes. The actual exam will
have different problems.
To best prepare for the exam, we recommend to study the lecture notes with
examples, DIY problems, assignment problems, and examples in tutorials.
Problem #1
(4pts)
Use the comparison test to decide the convergence of the integral
0
x
∫−∞ x(1 + cos(x))e dx.
Clearly state the function that you use for the comparison test.
1
Problem #2
(6pts) [2pts each]
Evaluate the following integrals:
a) ∫ (t2 − 3) ln(t) dt
2
x2
3
3
0 (8x +27) 2
b) ∫
c) ∫
1
(x2 +1)x
dx
dx
2
Problem #3
(4pts)
Use trigonometric substitution to evaluate ∫
3
√ 1
( 4−x2 )3
dx.
Problem #4
(4pts)
Find the volume of the solid obtained by rotating the region bounded by
y = 2x − x2 and y = 0 about the x-axis.
4
Problem #5
(5pts)
Consider the solid created by revolving the region bounded by the curves
y = x8 and the line y = 10−2x about the y-axis. (You may use that x8 = 10−2x
for x = 1 and x = 4.)
a) Draw the solid in the diagram below. Clearly label all curves. [1pt]
b) Find the volume of the solid using the disk/washer method. (Set up,
but do not integrate.) [2pts]
c) Find the volume of the solid using the cylindrical shell method. (Set
up, but do not integrate.) [2pts]
5
Problem #6
(3pts)
Find the general solution for x > 0 to the differential equation
x2 y ′ − xy = 3.
6
Problem #7
(5pts)
Consider the differential equation
y′
= (2yx)2 .
2
x
a) Classify the differential equation by stating its order and deciding if it is
linear, and state if it is separable. [1pt]
b) Find the general solution, y(x), to the differential equation.[3pt]
c) Find the particular solution that solves the differential equation with the
initial condition y(0) = 5.[1pt]
7