Solutions Examination 2 Calculus 2
1.
Find the volume generated by rotating the area between the curves y = 5 x and y = x2 for 0 x 3 about the x - axis.
HINT: washers
SOLUTION. The volume can be computed using washers. The cross sectional area is
AHxL = p
Solutions Examination 1 Calculus 2
1a. On the plot below shade the area given by the right Riemann sum for 0 sin x x for the partition 90,
2p
p
,
2
p,
3p
,
2
SOLUTION.
2 p=.
b. Consider the continuous function f HxL on [a, b]. Why is the definite integral
Solution Calculus 2 Examination 1
1.
What is a Riemann sum?
SOLUTION. Let f HxL be a real valued function defined on the interval I = @a, bD. Let 8x0 , x1 , , xn < be a partition of I , i.e., a
set of points with a = x0 < x1 < xn = b. Then a Riemann sum o
Calculus 2 Sample Examination 2
1.
Find the volume obtained by rotating the area between the graph f HyL =
1 + y3 for 1 y 2 and the y - axis about
the y - axis.
2
x=
1
!
1 + y3
1
2
3
SOLUTION. Using washers, we get
V = 1 p
2
2
2.
1 + y3
= pJy +
y = p 1 I
Solution Quiz D
Evaluate
1.
1
I1- x2 M
32
x.
SOLUTION. Let x = sin q; then I1 - x2 M
32
32
= I1 - sin2 qM
= Icos2 qM
32
positive since arcsin x = q lies in the interval A- 2 , 2 E. So we get
1
I1- x2 M
32
x =
cos q
cos3 q
using the fundamental triangle