base of a solid in the region bounded by parabola x^2+y=4 and x+y=2. Cross sections of the solid perpendicular to the x-axis are semicircles. what is the volume

Question

The base of a solid in the region bounded by the parabola x2 + y = 4 and the line x + y = 2. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

81 times pi over 20
81 times pi over 40
81 times pi over 80
108 times pi over 5

Thanks a lot!

Answer 1

Hi Lucy,

 

Please see the following graph to get the idea. 

 

https://www.desmos.com/calculator/iv7cppipo6

 

Some texts call this volume by "slicing". You need to find a formula for the area of a typical cross-section. If you solve the equation of the parabola for y, you get y = 4 - x². For the line you have y = 2 - x. The length of a diameter is given by:

(4 - x²) - (2 - x) = -x²+x+2. Take half of that to get the radius of a semi-circle. Substitute that into the formula for the area of a semi-circle: A = πr²/2. Then integrate with respect to x from x = -1 to x = 2 (those are the x coordinates of the points of intersection of the two curves that form the base of the solid.