The region between the graphs of x=y2 and x=5y is rotated around the line y=5. Find the volume of the resulting solid.
To find the volume of the resulting solid, we need to use the method of cylindrical shells. Here's the step-by-step process to find the volume:
Step 1: Visualize the region
Graph the given equations x = y^2 and x = 5y on a coordinate plane to see the region between them.
Step 2: Determine the limits of integration
To find the limits of integration, we need to determine the points of intersection between the two graphs. Set the equations equal to each other: y^2 = 5y. Rearrange to get y^2 - 5y = 0 and factor: y(y - 5) = 0. So, the points of intersection are y = 0 and y = 5.
Step 3: Set up the integral
The volume of the solid can be found by integrating the circumference of the cylindrical shells with respect to y.
We need to express the circumference of each shell as a function of y. Since the axis of rotation is y = 5, the distance from the axis of rotation to a point on the curve y = y is given by the expression r = 5 - y.
The circumference of the shell will be 2πr, and the height of the shell will be the difference between the x-values at each y, which is x = 5y - y^2.
Therefore, the integral for the volume is ∫(2π(5 - y)(5y - y^2))dy, where the limits of integration are from y = 0 to y = 5.
Step 4: Evaluate the integral
Integrate the expression formed in Step 3:
V = ∫(2π(5 - y)(5y - y^2))dy, with the limits of integration from 0 to 5.
After integrating, you will get the volume as a numerical result.
Step 5: Calculate the volume
Evaluate the integral and perform the necessary calculations to find the volume of the solid.
That's how you can find the volume of the resulting solid, using the method of cylindrical shells.