## Transcribed Text

Question 1:
(a) (8 pts) Estimate the area under the graph of f (x) = 9 x2 on the interval [3, 3] using 3
approximating rectangles and the right end points.
(b) (2 pts) Find an antiderivative of f(x) = 9 x2.
(c) (8 pts) Using part 2 of the Fundamental Theorem of Calculus find the area under the graph of f(x)=9x2 fromx=3tox=3.
2
Question 2:
(a) (8 pts) Express the Riemann sum
n!1 i=1 n n as a definite integral on the interval [1, 1].
(b) (8 pts) Express the definite integral
Z 12 px dx
as the limit of a Riemann sum.
Xn 2 ✓ 2i◆ lim sin 1+
5
3
Question 3:
(a) (8 pts) Use FTC part 1 to find the derivative of
g(x) =
tan t
p 3 dt
Z sinx
8 1+t
(b) (8 pts) Evaluate the indefinite integral
Z✓ 4 +sec2x+1◆dx
1+x2 x
4
Question 4:
(a) (8 pts) Use u-substitution to evaluate the following indefinite integral:
Z
4 sin(ln x) dx x
(b) (8 pts) Use integration by parts to evaluate the following indefinite integral:
Z
3xe8x dx
5
Question 5: Consider the region R bounded by x = 5 y2 and x = 14 y2.
(a) (6 pts) Graph x = 5 y2 and x = 14 y2, then identify the region bounded by the equations.
Clearly label the graph.
(b) (4 pts) For what values of y does 5y2 = 14y2?
y
(c) (6 pts) Find the area of the region R bounded by the functions x = 5 y2 and x = 14y2.
6
x
Question 6:
(a) (8 pts) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating
the region bounded by y = ex, y = 0, x = 0, and x = 1 about the x-axis.
(b) (8 pts) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = ex, y = 0, x = 0, and x = 1 about the line x = 2.
7

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.