Revolve the region bounded by y = 4x and y = x^2 about the y-axis. In cubic units, the resulting volume is

First you have to find where they intersect, that is easy

....
I am sure you can see that is (0,0) and (4,16)
The radius of the outer ring is √y, the radius of the inner ring is y/4
Vol = π∫(y - y^2/16) dy
= π[ y^2/2 - y^3/48] from 0 to 16
= π[128 - 256/3 - 0) = (128/3)π

you better check my arithmetic, getting sloppy in my old age.

Well, isn't that just a lovely mathematical problem? It's like asking a clown to juggle numbers and shapes!

To find the volume, we'll use the method of cylindrical shells. Since we're revolving the region bounded by y = 4x and y = x^2 about the y-axis, let's find the limits of integration first.

The two curves intersect at (0, 0) and (4, 16), so our limits of integration will be from 0 to 4.

Now, let's consider a vertical strip of width "dx" at a distance of "x" from the y-axis. When we revolve this strip about the y-axis, it forms a cylindrical shell.

The height of the shell will be the difference between the two curves, which is (4x - x^2). The circumference of the shell will be 2πx. The thickness of the shell is "dx". So, the volume of this shell is (2πx)(4x - x^2)(dx).

To find the total volume, we integrate this expression from x = 0 to x = 4:

V = ∫[0 to 4] (2πx)(4x - x^2) dx

Solving this integral will give us the resulting volume in cubic units. But hey, who needs all these calculations? Let's just appreciate the beauty of math and the wonderful shapes it creates!

To find the volume of the solid generated by revolving the region bounded by y = 4x and y = x^2 about the y-axis, we can use the method of cylindrical shells.

Step 1: Find the points of intersection of the two curves.
Set y = 4x equal to y = x^2:
4x = x^2
x^2 - 4x = 0
x(x - 4) = 0
x = 0 or x = 4

So the two curves intersect at x = 0 and x = 4.

Step 2: Determine the height of each cylindrical shell.
The height of each cylindrical shell is given by the difference between the two curves at a particular x-value. In this case, it is y = 4x - x^2.

Step 3: Determine the radius of each cylindrical shell.
The distance between the y-axis and a particular x-value is the radius of the cylindrical shell. In this case, it is given by r = x.

Step 4: Set up the integral to find the volume.
The volume of the solid is given by the integral:

V = ∫[a,b] 2πrh dx

where a and b are the x-values of the points of intersection.

In this case, a = 0 and b = 4. Therefore, the integral becomes:

V = ∫[0,4] 2πx(4x - x^2) dx

Step 5: Evaluate the integral.
V = 2π ∫[0,4] (4x^2 - x^3) dx
V = 2π [4/3 x^3 - 1/4 x^4] evaluated from 0 to 4
V = 2π [(4/3 (4)^3 - 1/4 (4)^4) - (4/3 (0)^3 - 1/4 (0)^4)]
V = 2π [(4/3 (64) - 1/4 (256) - 0]
V = 2π (256/3 - 64)
V = 2π (832/3)

Therefore, the resulting volume is 832/3 π cubic units.

To find the resulting volume when the region bounded by the equations y = 4x and y = x^2 is revolved about the y-axis, we need to use the method of cylindrical shells.

First, let's find the intersection points of the two curves:

4x = x^2

Rearranging, we get:

x^2 - 4x = 0

Factorizing, we have:

x(x - 4) = 0

So, x = 0 or x = 4. These are the x-coordinates of the points where the two curves intersect.

To set up the integral for the volume, we need to consider a small element of height dy and a representative shell of thickness dx at height y.

Now, let's express the equations in terms of x:

y = 4x => x = y/4
y = x^2 => x = sqrt(y)

The representative shell has a height of dy and its radius is the distance between the curves at height y, which is x = sqrt(y) - y/4.

The volume of each shell can be considered as a cylinder with circumference 2π times the radius and height dy. So, the volume of each shell is given by:

dV = 2π(radius) * dy
= 2π(sqrt(y) - y/4) * dy

Integrating this expression over the interval from y = 0 to y = 16 (the intersection points) will yield the total volume:

V = ∫[0, 16] (2π(sqrt(y) - y/4)) dy

Evaluating this integral will give the resulting volume in cubic units.