Question

1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = ln(5x), y = 1, y = 3, x = 0; about the y-axis
2. Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = 32 − x2, y = x2; about x = 4
3. Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8.
64y = x3, y = 0, x = 8
4. A tank is full of water. Find the work required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft3. (Assume a = 6 ft, b = 8 ft, and c = 12 ft.)
5.If 2.4 J of work are needed to stretch a spring from 9 cm to 13 cm and 4 J are needed to stretch it from 13 cm to 17 cm, what is the natural length of the spring?

Answer 1

another homework dump?
Looks like you need to show some work. I'll get you started.
#1. y = ln(5x), y = 1, y = 3, x = 0; about the y-axis
using shells of thickness dx,
v = 2(e/5) + ∫[e/5, (e/5)^3] 2πrh dx
where r = x and h=3-ln(5x)

using discs of thickness dy,
v = ∫[1,3] πr^2 dy
where r = x = e^(y/5)

#2.
v = ∫[-4,4] 2πrh dx
where r = 4-x and h = 32-x^2 - x^2

#3.
v = ∫[0,8] 2πrh dy
where r = 8-y and h = x = 4∛y

#4.
No idea what a,b,c are supposed to be. But, find where the center of mass is for the tank, and the work required is just weight * height_lifted
and the weight is the volume * density

#5.
work = 1/2 kx^2
where x is the distance stretched.
So, if the natural length is a, we have
k/2 ((13-a)^2 - (9-a)^2) = 2.4
k/2 ((17-a)^2 - (13-a)^2) = 4
Now divide and solve for a.

Answer 2

Oops. I'm sure you caught my error in #1.
v = π(e/5)^2(2 + ∫[e/5, (e/5)^3] 2πrh dx