Let f and g be the functions given by f(x)=1+sin(2x) and g(x)=e^(x/2). Let R be the shaded region in the first quadrant enclosed by the graphs of f and g.

Question

A. The region R is the base of a solid. For this solid, the cross sections, perpendicular to the spx-axis, are rectangles with height 5. Find the volume of this solid.

B. The region R is the base of a solid. For this solid, the cross sections, perpendicular to the x-axis, are squares. Find the volume of this solid.

Answer 1

the graphs intersect at (0,1) and (1.136,1.764)

A
The cross-sections have base of f(x)-g(x) and height 5. So, we add them all up:

v = ∫[0,1.136] 5(1+sin(2x)-e^(x/2)) dx
= 5(x - 1/2 cos(2x) - 2e^(x/2)) [0,1.136]
= 2.1455

B
The same thing, but the height is the same as the base, so

v = ∫[0,1.136] ((1+sin(2x)-e^(x/2))^2 dx

you'll have to use integration by parts to do the sin(2x)*e^(x/2) stuff.

If you scroll down a bit here, you can see the final indefinite integral:

http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0%2C1.136%5D+%28%281%2Bsin%282x%29-e%5E%28x%2F2%29%29%5E2+dx

Answer 2

Here was my first answer
http://www.jiskha.com/display.cgi?id=1429478661

In a subsequent reply you said you were confused.

Now you simply repost the same question for a third time. Exactly what did you not understand in my answer?

Answer 3

I replied to those ones again