Use the disc method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.

y=x^3, y=0, x=2
a)the x-axis
b)the line x=4

we've done several of these for you. Got any ideas on this one?

To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about a given line, we'll use the disc method. The disc method involves summing up the volumes of infinitely many thin discs that are formed as the region is rotated.

a) Revolving about the x-axis:
When the region is revolved about the x-axis, the resulting solid is symmetric about the x-axis. This means we can consider a representative disc at any point on the x-axis and multiply its volume by the appropriate factor to obtain the total volume.

To find the volume of a representative disc, we need to consider a thin vertical strip of the region. The cross-section of this strip perpendicular to the x-axis is a disc with radius y (since the width of the strip is dy) and thickness dx.

The volume of an infinitesimally thin disc is given by V = π(radius)^2(thickness). In this case, the radius is y, which is equal to x^3, and the thickness is dx.

So, the volume of the representative disc is dV = π(x^3)^2(dx). Summing up the volumes of all such discs from x = 0 to x = 2 will give us the total volume:

V = ∫[0 to 2] π(x^3)^2(dx)
= ∫[0 to 2] πx^6 dx

Evaluating the integral, we get:

V = π[1/7(x^7)] [0 to 2]
= π/7(2^7 - 0)
= 128π/7
≈ 57.82 cubic units.

Therefore, the volume of the solid generated when the region bounded by the graphs y = x^3, y = 0, and x = 2 is revolved about the x-axis is approximately 57.82 cubic units.

b) Revolving about the line x = 4:
When the region is revolved about the line x = 4, the resulting solid is symmetric about the line x = 4. Similarly to part a), we can consider a representative disc at any point on the line x = 4.

To find the volume of a representative disc, we need to consider a thin vertical strip of the region. The cross-section of this strip perpendicular to the line x = 4 is a disc with radius equal to the distance between the line x = 4 and the graph y = x^3.

The distance between the line x = 4 and the graph y = x^3 is given by the difference between x and 4, i.e., (x - 4). Therefore, the radius of the representative disc is (x - 4), and the thickness is dx.

The volume of the representative disc is given by dV = π(radius)^2(thickness). Substituting the values, we get:

dV = π(x - 4)^2(dx).

Summing up the volumes of all such discs from x = 0 to x = 2 gives us the total volume:

V = ∫[0 to 2] π(x - 4)^2(dx).

Evaluating the integral, we get:

V = π[1/3(x - 4)^3] [0 to 2]
= π/3[(2 - 4)^3 - (0 - 4)^3]
= π/3[(-2)^3 - (-4)^3]
= -π/3(8 - 64)
= -56π/3
≈ -59.05 cubic units.

Note that the negative sign indicates that the volume is below the x-axis. Therefore, the volume of the solid generated when the region bounded by the graphs y = x^3, y = 0, and x = 2 is revolved about the line x = 4 is approximately -59.05 cubic units.