Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 1−y^4, x = 0; about x = 1.
using shells of thickness dx, and taking advantage of symmetry,
v = 2∫[0,1] 2πrh dx
where r=1-x and h=y=∜(1-x)
v = 2∫[0,1] 2π(1-x)∜(1-x) dx
= 4π∫[0,1] (1-x)^(5/4) dx = 16π/9
using discs (washers) of thickness dy, we have
v = 2∫[0,1] π(R^2-r^2) dy
where R=1 and r=1-x
v = 2∫[0,1] π(1-(1-(1-y^4))^2) dy
= 2∫[0,1] π(1-y^8) dy
= 16π/9
To find the volume of the solid obtained by rotating the region bounded by the curves about the line x = 1, we can use the method of cylindrical shells.
First, let's sketch the given curves and the rotation axis x = 1 on a graph to visualize the region of interest.
The curves given are x = 1 - y^4 and x = 0. The curve x = 1 - y^4 is a fourth-degree polynomial that opens towards the right. The curve x = 0 is simply the y-axis.
Now, to set up the integral for finding the volume, we need to determine the limits of integration and the height of each cylindrical shell.
1. Determining the limits of integration:
Since x = 0 is the left boundary curve and x = 1 - y^4 is the right boundary curve, we need to find the intersection points of these two curves to determine the limits of integration for y.
Setting x = 1 - y^4 equal to x = 0, we get:
1 - y^4 = 0
y^4 = 1
y = ±1
Therefore, the limits of integration for y are y = -1 to y = 1.
2. Determining the height of each cylindrical shell:
The height of each cylindrical shell is the difference between the x-coordinate values of the curves at a given y-value. In this case, the height is the difference between the x-coordinate of the curve x = 1 - y^4 and the x-coordinate of the line x = 1 at a given y-value.
For a given y, the height of the cylindrical shell is:
h = (1 - y^4) - 1
h = 1 - y^4 - 1
h = -y^4
Now, we can set up the integral to find the volume:
V = ∫[from y = -1 to y = 1] of 2πrh dy
= ∫[from y = -1 to y = 1] of 2π(-y^4)dy
= 2π ∫[from y = -1 to y = 1] of -y^4 dy
Evaluating this integral will give us the volume of the solid.
I hope this explanation helps you understand how to find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.