Calculus Washer Method?

Question

A cylindrical hole of radius 3 is bored horizontally, exactly through the center of a solid sphere of radius 5. Use the washer method to find the volume of the remaining portion of the sphere.

I took the inner radius of the washer to be 3, and the outer radius to be 5. Doing it this way I get the area to be:

pi(5)^2 - pi(3)^2 = 16pi

However there is no formula there to set up an integral to get the volume, so I must be doing something wrong but I'm not sure what.

I'm in calculus 1 so we haven't done really anything with polar coordinates, but we have learned how to change the limits of integration when using the substitution method.

Answer 1

Do you know how to use polar or spherical coordinates? If so, then set up the integral for the full sphere and adjust the limits.

Answer 2

draw the equations on the graph. By drawing them, you see that they intersect at two points. Next you need to find where they intersect. To do that you let them be equal to each other. 5x+2=3x^2 rewrite it 0=-2-5x+3x^2 or 3x^2-5x-2=0 its a quadratic equation so you factor to solve for x (3x+1)(x-2)=0 x-2=0,3x+1=0 so both function intersect at x=2,x=-1/3 put the numbers back into the equation to see if you get the same number for both. If you do then these are you limits. If you are rotating about the x-axis, then these are your limits, if you rotating about the y-axis, then put these numbers in to get your y components. V=intergral(pi[R(x)]^2-[r(x)]^2)dx by looking at the graph you see that y=5x+2 is the top function and y=3x^2 is you bottom function so V=int[pi{5x+2}^2-{3x^2)^2]dx from -1/3 to 2.