I want to really know this. Consider the solid obtained by rotating the region bounded by the given curves about the line x = 4. x= 4y^2 , x = 4

Question

What is the radius, cross-sectional area, intersection of points, and volume? I gladly appreciate whoever can help. Thanks. 

Answer 1

Hi Ab, 

 

This would be easier for you to see with a picture. The x=4y² is a parabola opening to the right with vertex at (0,0). Since we are revolving about the line x=4, picture a typical radius emanating from that line extending to the parabola. What is the length of that segment?  The expression representing it is 4 - 4y². 

 

At the risk of having Wyzant un-approve this answer because of an embedded link, here is what it looks like:

 

https://www.desmos.com/calculator/o0lkypbmke  (the graph only shows the top 1/2 of the parabola)

 

If you picture the blue "radius" revolving around the line x = 4 you should picture a circle with radius (4-4y²). So the area of that circle will be π(4-4y²)². You need to know where x=4 intersects the parabola so that you can determine the limits of integration for calculating the volume. Let x=4 and solve for y. When x = 4, y = +/- 1.

 

So to calculate the volume of revolution integrate along the y-axis (the axis parallel to the line you are revolving about). In a situation like this I usually integrate from 0 to 1, then double the answer (but from -1 to 1 works too).

 

So,  π∫ (4-4y²)² dy from -1 to 1 (or from 0 to 1, then double your answer).