Calculus Volumes?

Question

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

Find the volume of the solid generated when R is revolbed about the horizontal line y = 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.

Answer 1

First we have to find the intercepts
e^-3x=sqrt(x)
Its easier taking log
-3x=1/2lnx
so take the function y= lnx+6x
limy x=>0+=-infinity
y´=1/x+6 >0 for x>0 The function is increasing and has only one root between 0 and 1
x=0,24 is an approximate value

The volume is the integral between 0,24 and 1 of
pi[(1-e^-3x)^2-(1-sqrt(x))^2]dx You can calculate it

The volume is Int between 0,24 and 1 of 5(sqrtx-e^-3x)^2dx