Finding the volume of a region with a known cross section.

[tarynt1]

Let R be the shaded region bounded by the graphs of y = sqrt(x) and y = e^(-3x) and the vertical line x = 1.

(Since you can't see the diagram, R is the largest enclosed region directly to the left of the line x=1.)

The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.

Again, all I need help with is setting up the integral. Thanks!

 

[tkhunny]

Setting up the integral is the part you are supposd ot be learning.

Find the intersection \(\displaystyle \sqrt{x} = e^{-3x}\).

I get about x = 0.2382

Decide which is greater over your region.

Really, that's all the tricky parts. Finding the point of intersection is the only real challenge.