MATH 101 Quiz #1 (v.T1)
Last Name:
Thursday, January 14
First Name:
Grade:
Student-No:
Section:
Very short answer question
1. 1 mark If
Z
2
f (x) dx = 4 and
0
Z
2
g(x) dx = 3, calculate
0
your answer in the box. Simplify your answer completely.
Z
2
3f (x)
2g(x) dx. Please write
0
Answer:
Short answer questions—you must show your work
2. 2 marks Use elementary geometry to calculate
f (x) =
(
Z
3
f (x) dx, where
0
x, if x  1,
1, if x > 1.
3. 2 marks The value of the following limit is equal to the area below a graph of y = f (x),
integrated over the interval [0, b]:
 ✓
◆
n
X
4
4i
lim
cos 2 +
n!1
n
n
i=1
2
.
Find f (x) and b. (Do not evaluate the limit.) Please write your answers in the boxes.
Answer:
Answer:
Long answer question—you must show your work
4. 5 marks Consider the integral:
Z
5
(5 + x2 ) dx.
(⇤)
0
(a) (1 mark) Approximate this integral using the left Riemann sum with n = 5 intervals.
(b) (4 marks) Write down the expression for the right Riemann sum with n intervals and
calculate the sum. Now take the limit n ! 1 in your expression for the Riemann sum,
to evaluate the integral (⇤) exactly. You can leave your answer in “calculator-ready” form.
Hint: you may use the identity
n
X
i=1
i2 =
2n3 + 3n2 + n
.
6