CALCULATOR PART
1. The area of R + S is unsigned, meaning you want to find

where
is the interval between the leftmost and rightmost intersections of
and
.
First use your calculator to find these intersections:

so that
and
. Now compute the integral using your calculator:

2. The volume, using the washer method, is given by the integral

3. A circle of radius
has area
; a semicircle with the same radius thus has area
. Each cross section of this solid is a semicircle whose diameter is the vertical distance between
and
, or
. In terms of the diameter
, the area of each semicircle would be
. Then the volume of the solid is

NON-CALCULATOR PART
4. The mean value theorem says that for a function
continuous on an interval
and differentiable on
, there is some
such that

If this
happens to be an antiderivative of
, then we end up with

is continuous and differentiable everywhere, so the MVT applies. We have
, so the MVT tells us there is some
such that

That is, the average value of
on
is 0. The MVT says there is some
in the interval such that the function takes on the average value itself; this happens for
.
5. This question seems to be incomplete...
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