Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
x=9y^2 y=1 x=0 about the y-axis.
I am able to find the volume of other questions, but this one got me stumped. I tried drawing out the graph, but I can't really form the boundaries. help? thank you
To find the volume of the solid obtained by rotating the region bounded by the curves x = 9y^2, y = 1, and x = 0 about the y-axis, you can use the method of cylindrical shells.
1. First, let's sketch the region bounded by the given curves.
- Since x = 9y^2, we can rewrite it as y = √(x/9).
- The curve y = 1 is a horizontal line passing through y = 1.
- The curve x = 0 is the y-axis.
Plotting these curves, you will have a parabolic shape that opens towards the right, bounded by y = 1 at the top and the y-axis at the bottom.
2. Now, let's find the limits of integration for the integral that calculates the volume.
- The region is bounded by y = 1, so we will integrate with respect to y.
- The lower limit (y-value) is at the intersection of the parabola and the line y = 1.
By substituting y = 1 into the equation of the parabola, we get x = 9(1)^2 = 9.
So, the lower limit is y = 1 and the upper limit is y = 0.
3. Setup and evaluate the integral for the volume using cylindrical shells.
- The formula for the volume using cylindrical shells is V = 2π ∫[a,b] x * h(y) * dy, where a and b are the limits of integration, x is the radius of the shell, and h(y) is the height of the shell.
- The radius of each shell is x, which is equal to the value of y at any given point since we are rotating the region about the y-axis.
- The height of each shell is the difference between the upper and lower bounds of the parabolic shape which is 1-0 = 1.
- The integral for the volume becomes V = 2π ∫[0,1] y * 1 * dy.
4. Evaluate the integral.
- Integrating y with respect to y gives you ∫ y * dy = 1/2 * y^2.
- Applying the limits of integration, the volume formula becomes:
V = 2π ∫[0,1] y * 1 * dy = 2π ∫[0,1] y * dy = 2π * 1/2 * [y^2] [0,1] = π * [1^2 - 0^2] = π.
Therefore, the volume of the solid obtained by rotating the region bounded by the curves x = 9y^2, y = 1, and x = 0 about the y-axis is π cubic units.