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 solid formed by rotating the region between the graphs of x=y^2 and x=5y around the line y=5, we can use the method of cylindrical shells.
First, let's find the points of intersection between the two curves by setting them equal to each other:
y^2 = 5y
Bringing all terms to one side, we get:
y^2 - 5y = 0
Factoring out the common factor y, we have:
y(y - 5) = 0
So, we have two possible values for y: y = 0 and y = 5.
Now, let's set up the integral that will give us the volume. Since we are using cylindrical shells, the differential volume element will be a thin strip with height dx, width of the circumference of the shell, and length of the difference in y-coordinates between the curves at x.
The radius of each shell is given by the distance between the line y = 5 and the curve x = y^2, which is 5 - y^2.
The height of each shell is given by the difference in x-coordinates between the curves, which is 5y - y^2.
The volume of each shell is the product of the height, circumference, and width:
dV = 2π(5 - y^2)(5y - y^2) dx
To find the total volume, we integrate this expression over the interval where the curves intersect, which is from y = 0 to y = 5:
V = ∫[0 to 5] 2π(5 - y^2)(5y - y^2) dx
Now, we need to express the limits of integration in terms of x. To do this, we solve the equations y = 0 and y = 5 for x:
x = y^2 ---> x = 0
x = 5y ---> x = 5(5) = 25
Therefore, the limits of integration become from x = 0 to x = 25:
V = ∫[0 to 25] 2π(5 - y^2)(5y - y^2) dx
Finally, we can proceed to evaluate the integral and find the volume of the resulting solid.