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1. (5 points) Evaluate the indefinite integral
Note: This is a multi step problem . You may use subsitution and then find out
you need
apply another method to the new integral.
2. (5 points) Consider the region bounded by the curves y = cos x, y = 1, x =
0, and
x = π/2 and the solid generated by rotation the region
about the x axis. Write as a
definite integral the expression for the volume V of the solid by (i) using the
washer
method and (ii) using the shell method. Evaluate exactly one of these integrals.
Note:
Your answer should have two (2) integrals, one evaluated and one unevaluated.
You
might need the formula
Solution: First, we set up the integral for volumes using the
shell method. For this,
the outer radius is 1 and the inner radius is cos x. Thus, we have the
volume is
Next, we use the shell method, but we need to get the functions in
terms of x . The
curves that bound the region are x = π/2 and x
= arcsin y. So, we get the volume is
The first integral is solved by integrating sin2x and the second
integral is solved
using parts. The final answer is |
3. (5 points) Let m and n be positive integers and prove
that
Solution: Here, we just use the
substitution u = 1 - x which gives du = -dx.
Observe if x = 0, then u = 1; also, if x = 1, then u = 0. Finally, we
note that
u = 1 - x implies that x = 1 - u, thus
Replacing u by x gives the result. (Note: We can
do this because u is just a dummy
variable in the previous expression . If that doesn't convince you, then
do a new
substitution u = x which gives du = dx and convice yourself that way.) |
4. (5 points) Evaluate the fol lowing inde nite integral
Solution: On the first integral we use
parts. We let  and dv = xdx which
implies  . So, we get
For the second part of the integral, we take out a
sin x from the expression and
replace the rest with cosines. So, we have
|
5. (5 points) Find the volume of the ellipsoid obtained by
rotating the region bounded by
the curve 
and the x axis, about the x axis.
Solution: Probably the easiest way to
solve this problem is to use the shell method.
Here, we only have one function bound and so the volume is
 |
