Finding volume of region with cross sectionsThe base of a solid is the region is between the line of y = 4 and the parabola y = x 2 the cross sections of the solid that are perpendicular to the x-axis are semicircles. What is the volume?So the question I have on this is whether or not I'm going to need to divide the integral by 2 since they are semi circles, not circles. When I divide by 2, I get 16 π 3 ; however, when I don't divide, I get 512 π 15 . Are either of these correct, and if so, which one?

Question

Finding volume of region with cross sectionsThe base of a solid is the region is between the line of   y  =  4 and the parabola   y  =      x    2   the cross sections of the solid that are perpendicular to the x-axis are semicircles. What is the volume?So the question I have on this is whether or not I&#039;m going to need to divide the integral by 2 since they are semi circles, not circles. When I divide by 2, I get             16              π              3  ; however, when I don&#039;t divide, I get             512              π              15  . Are either of these correct, and if so, which one?

Keely Blankenship · Accepted Answer

Explanation:What is one way to look at the area of the base of the solid? Here is one:      ∫          y      =      0              2        2  x     d  y  .We&#039;re cutting it into horizontal slices with length 2x and height dy.Similarly what function of x would describe the area of the semicircle?Then put that function instead of 2x in the formula for the area of the base of the solid.

Region is between the line of y=4 and the parabola y=x^2 the cross sections of the solid that are perpendicular to the x-axis are semicircles.

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