Math 2250 HW #13
Due 12:30 PM Thursday, November 21
Reading: Hass §5.4–5.6
Problems: Do the assignment “HW15” on WebWork. In addition, write up solutions to the
following problems and hand in your solutions in class on Thursday.
1. True or False: If f (x) and g(x) are integrable functions on the interval [a, b], then
Z b
Z b
Z b
f (x)g(x) dx =
f (x) dx
g(x) dx .
a
a
a
If your answer is “true”, explain why. If your answer is “false”, give a counterexample.
2. Find the shaded area:
6
y ‡ x + sinHxL
4
2
Π
2Π
3. If the function f (x) is integrable on the interval [a, b], then the average value of f (x) on [a, b]
is defined to be
Z b
1
f (x) dx.
b−a a
(a) Show that this definition of average value gives the value you would expect for the
function f (x) = 2x on the interval [1, 3].
(b) The points (cos(θ), sin(θ)) as θ ranges from −π/2 to π/2 trace out a semicircular arc.
1
à
HcosHΘL, sinHΘLL
1
2
- 12
1
2
1
- 12
-1
In particular, notice that the function x(θ) = cos(θ) gives the x-coordinates of the points
on this semicircle. What is the average x-coordinate of a point on this semicircle? (Note:
the answer is not 1/2.)
1