# Calculus II

posted by .

Find the volume of the solid generated by revolving R about the x-axis where R is the region enclosed by the larger curve y=(e^2x)/3, the smaller curve y=1 and the line x=ln(3)

• Calculus II -

intersection of y = (1/3)e^(2x) and y = 1
e^2x = 3
2x = ln3
x = (1/2)ln 3
so let's take the volume of the whole region below (1/3)e^(2x) from x = (1/2)ln3 to ln3 and subtract the small cylinder

Vol = π∫y^2 dx - inside small cylinder
= π∫(1/9)e^(4x) dx - i.s.c.
=π[(1/36)e^(4x) from (1/2)ln3 to ln3 - i.s.c.
= π[( (1/9)(81) - (1/9)(9) ) - i.s.c.
= π(9-1) - i.s.c.
= 8π - inside small cylinder

the inside small cylinder has a radius of 1 (from y=1) and a height of ln3 - (1/2)ln3 = (1/2)ln3
its volume is π(1^2)(1/2)ln3
= πln3 /2

whole volume = 8π - (1/2)(π)(ln3) or appr 23.4

I am pretty sure of my method, but you better check my arithmetic and calculations.

• Calculus II -

Could you show how to do this problem using integrals?

## Similar Questions

1. ### calculus

if the region enclosed by the y-axis, the line y=2 and the curve y=the square root of x is revolved about the y-axis, the volume of the solid generated is?
2. ### calculus

Find the volume of the solid generated by revolving the region about the given line. The region is in the first quadrant bounded above by the line y= sqrt 2, below by the curve y=secxtanx, and on the left by the y-axis. Rotate the …
3. ### Calculus 2

The volume of the solid generated by revolving the region about the given line. above the curve y=3 below the curve y=sqrt of 3x on the left by the y-axis the line y=3
4. ### Calculus

Find the volume of the solid generated by revolving the following region about the given axis. The region in the first quadrant bounded by the curve y=x^2, below by the x-axis, and on the right by the line x=1, about the line x=-2
5. ### Calculus AB...I really need help

The region in the first quadrant enclosed by the coordinates axes, the line x=pi, and the curve y= cos(cosx) is rotated about the x-axis. What is the volume of the solid generated.
6. ### Math

Find the volume of the solid generated by revolving the following region about the given axis The region in the first quadrant bounded above by the curve y=x^2, below by the x-axis and on the right by the line x=1, about the line x=-4
7. ### calculas

find the volume of the solid generated by revolving the region about the given line. the region in the first quadrant bound above by the line y=1, below by the curve y=√(sin5x) ,and on the left by the y-axis, above the line y=-1
8. ### calculus

Find the volume of the solid generated by revolving the region about the given line. The region in the second quadrant bounded above by the curve y = 16 - x2, below by the x-axis, and on the right by the y-axis, about the line x = …
9. ### cal

Solve the problem. Find the volume of the solid generated by revolving the region bounded by the curve y=lnx, the x-axis, and the vertical line x=e^(2) about the x-axis.
10. ### Calculus

Use the disk method to find the volume of the solid generated by revolving about the y-axis the region bounded by the curves y=8−x^2 and the curve y=x^2.

More Similar Questions