Find the volume of the solid obtained by rotating the region bounded by

y=8x+24, y=0
about the y-axis.

I assume that the line x=0 is also part of the boundary. If so, you just have a cone with radius 24 and height 3, with a volume of

v = π/3 * 24^2 * 3 = 576π

Oh, ok -- if you want to use calculus, then using discs of thickness dx,

v = ∫[-3,0] πr^2 dx
where r = y = 8x+24
v = π∫[-3,0] (8x+24)^2 dx
= 64π∫[-3,0] (x+3)^2 dx
= 64π∫[-3,0] x^2+6x+9 dx
= 64π (1/3 x^3 + 3x^2 + 9x) [-3,0]
= 576π

using shells of thickness dy,

v = ∫[0,24] 2πrh dy
where r = y and h = x = -(y-24)/8
v = π/4 ∫[0,24] -y(y-24) dy
= -π/4 (1/3 y^3 - 12y^2) [0,24]
= 576π

Rats. I did it about the x-axis. Still, you see the method. Redo the rotation, and see whether you come up with 72π

To find the volume of the solid obtained by rotating the region bounded by the curves y = 8x + 24 and y = 0 about the y-axis, we can use the method of cylindrical shells.

First, let's find the points of intersection between the two curves:
Setting y = 8x + 24 equal to zero, we get:
0 = 8x + 24
Solving for x, we get:
x = -3

So the region bounded by the curves is from x = -3 to x = 0.

Now, let's set up the integral to find the volume using the formula for the volume of a solid of revolution by cylindrical shells:
V = 2π∫[a,b] x * h(x) dx

Where:
a = -3 (the lower limit of integration)
b = 0 (the upper limit of integration)
h(x) = 8x + 24 (the height of the cylindrical shell)
x = variable of integration

To evaluate this integral, we need to express x as a function of y since we are rotating the region about the y-axis. Rearranging the equation for x, we get:
x = (y - 24) / 8

Now, substitute x with (y - 24) / 8 in the integral:

V = 2π∫[-3,0] (y - 24)/8 * (8x + 24) dx

Simplifying, we have:

V = 2π∫[-3,0] (y - 24) * (x + 3) dx

Now, integrate from x = -3 to x = 0:

V = 2π∫[-3,0] (y - 24) * (x + 3) dx
= 2π * [(y - 24) * (x + 3)] |[-3,0]
= 2π * [(y - 24) * (0 + 3) - (y - 24) * (-3 + 3)]
= 2π * [(y - 24) * 3]
= 6π * (y - 24)

Now, the volume is expressed as a function of y. To find the limits of integration for y, let's find the value of y when the curves intersect. From the equation y = 8x + 24, substitute x = -3:

y = 8(-3) + 24
y = 0

So, the limits of integration for y are from y = 0 to y = 24.

Finally, integrate the expression for the volume with respect to y from y = 0 to y = 24:

V = ∫[0,24] 6π * (y - 24) dy

Evaluating this integral:

V = 6π∫[0,24] (y - 24) dy
= 6π * [((y^2)/2) - 24y] |[0,24]
= 6π * (288 - 288)
= 0

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = 8x + 24 and y = 0 about the y-axis is zero.