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FINAL EXAM
CALCULUS 2

MATH 2300
FALL 2018



Name
PRACTICE EXAM
SOLUTIONS




Please answer all of the questions, and show your work.
You must explain your answers to get credit.
You will be graded on the clarity of your exposition!




Date: December 12, 2018.
1

, 1

10 points
1. Consider the region bounded by the graphs of f ( x ) = x2 + 1 and g( x ) = 3 − x2 .

1.(a). (5 points) Write the integral for the volume of the solid of revolution obtained by
rotating this region about the x-axis. Do not evaluate the integral.



SOLUTION: We can see the region in question below.
y
3 f ( x ) = x2 + 1

2
1 g( x ) = 3 − x2
x
−1 1
Using the washer method, the volume integral is
Z 1 Z 1
π g( x )2 − f ( x )2 dx = π (3 − x2 )2 − ( x2 + 1)2 dx.
−1 −1




1.(b). (5 points) Write the integral for the volume of the solid of revolution obtained by
rotating this region about the line x = 3. Do not evaluate the integral.



SOLUTION: Now using the shell method, the integral is equal to
Z 1 Z 1
2π (3 − x )( g( x ) − f ( x )) dx = 2π (3 − x )((3 − x2 ) − ( x2 + 1)) dx
−1 −1
Z 1
= 2π (3 − x )(2 − 2x2 ) dx
−1




2