Consider the solid obtained by rotating the region bounded by the given curves about the line x = 4.
x= 4y^2 , x = 4
Find the volume V of this solid.
To find the volume V of the solid obtained by rotating the region bounded by the given curves about the line x = 4, we can use the method of cylindrical shells.
First, let's graph the curves x = 4y^2 and x = 4 to visualize the region that we'll be rotating.
The curve x = 4y^2 is a parabola that opens rightward and passes through the point (4, 1) on the y-axis. The curve x = 4 is a vertical line passing through the point (4, 0) on the x-axis.
Now, let's find the bounds of integration for the volume calculation. The region is bounded by the curves x = 4y^2 and x = 4. To find the bounds, we need to set up an equation for y in terms of x for each curve.
For x = 4y^2, we can solve for y by taking the square root of both sides and considering the positive and negative root:
y = ±√(x/4)
For x = 4, y is simply 0 since the line is parallel to the y-axis.
Since we are rotating about the line x = 4, the height of each shell will be the difference between the upper curve (x = 4y^2) and the lower curve (x = 4), which is given by:
h = (±√(x/4)) - 0 = ±√(x/4)
Now, let's set up the integral for the volume V using the cylindrical shells method:
V = ∫[a,b] 2πrh dx
Where r is the distance from the axis of rotation (x = 4) to the shell (which is simply x - 4), and h is the height of the shell.
Substituting the values for r and h, we have:
V = ∫[a,b] 2π(x - 4)(±√(x/4)) dx
Now, let's find the bounds of integration for x. Since the curves intersect at (4, 0), we can set up the integral from x = 0 to x = 4:
V = ∫[0,4] 2π(x - 4)(√(x/4)) dx
Next, we simplify and evaluate the integral to find the volume V of the solid.