The exact value of the volume of the object we obtain when rotating R about the x-axis is 128π/3.
To find the volume of the solid of revolution obtained by rotating the region R about the x-axis, we can use the disc method. This method involves slicing the solid into thin discs, each perpendicular to the axis of rotation, and then summing the volumes of these discs.
The disc method formula is:
V = ∫π(r(x))^2 dx
where:
V is the volume of the solid
π is pi
r(x) is the radius of the disc as a function of x
dx is the width of each disc
In this case, the radius of the disc as a function of x is equal to the distance between the curve y = x + 2 and the x-axis. So, r(x) = x + 2.
The interval of integration is [0, 4], since the region R starts at x = 0 and ends at x = 4.
Plugging into the disc method formula, we get:
V = ∫π(x + 2)^2 dx
V = π∫(x^2 + 4x + 4) dx
V = π(x^3/3 + 2x^2 + 4x) | from 0 to 4
V = π(64/3 + 32 + 16)
V = 128π/3
Therefore, the exact value of the volume of the object we obtain when rotating R about the x-axis is 128π/3.
Question
Suppose that R is the finite region bounded by y =x , y=x + 2, x = 0, and x = 4.Find the exact value of the volume of the object we obtain when rotating R about the x -axis.