Answer to Question #20450 in Integral Calculus for Darya

Question

These problems involve the use of integrals.

1. Let R be the region bounded by the graphs of y = √x and y= e^(-3x) and x=1
a) Find the area of region R
b) Find the volume of the solid generated when R is revolved around the x-axis
c) 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 6 times the length of its base in region R. Find the volume of this solid.

2. A solid lies between two planes perpendicular to the x-axis at x=1 and x=2. The cross-sections perpendicular to the x-axis are rectangles that have the bases running from the curve y=x^3/2 down to the x-axis. The height of the rectangle is twice the base.
Find an integral for the volume of the solid and solve analytically.




3. Find the volume of the solid that is formed by equilateral triangles with a base that lies perpendicular to the x-axis between y=3x^2+2 and y=3x+8.
(The area of an equilateral triangle is A=s^2 √3/4)