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A closed-notes and closed-book calculus exam from math 121's calculus ii course during spring 2005. The exam covers various topics such as integration, average values, sketching regions, and finding volumes of rotation. Students are required to evaluate integrals, find the average value of a function, sketch regions, and compute areas and volumes.
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Test #2 Name: Section 1 2 3 Instructor:
This exam is CLOSED NOTES and CLOSED BOOK. There are NO CALCULATORS allowed. To get full credit you must show all work neatly in the space provided on the test paper.
a.
∫ (^) s
(1 + s^2 )^3
ds
b.
∫ sin(x) cos^2 (x) dx.
c.
∫ (^1)
0
x^2 (1 − x^3 )^1 /^3 dx
d.
∫ (^3) − u
u
du
6
y
x
g(x) f (x)
a. Express the area between f (x), g(x) and y = 6 as a definite integral (or integrals) with respect to x.
b. Express the same area as a definite integral (or integrals) with respect to y.
5.(8 pts.) Consider the solid object pictured below. Suppose that the cross section of the object at height z is a square with diagonal 3z/2. What is the volume of the object.
¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢
6
° °™°
z
!!
!!! HH HH HH HH !! !!!
≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠
!!
!!
!!!HHH HHH !! !! HHH!!!
HHH
c. The line x = 3.
-2 -1 1
y 6
3
x
d. The line y = 4.
-2 -1 1
y 6
3
x
