AP Calc AB Problem - Finding volumeThe base of a solid in the region bounded by the two parabolas y 2 = 8 x and x 2 = 8 y. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?The answer choices are:A) 288 π 35 B) 576 π 35 C) 144 π 35 D) 8 πI started off by writing the integral like this: ∫ 0 8 1 2 π ( 1 2 ( 8 x − x 2 8 ) ) 2 d xthen I simplified it to π 8 ∫ 0 8 ( 8 x − x 2 8 ) 2 d xI can't think of anyway to solve this problem from here, without using a calculator.

Question

AP Calc AB Problem - Finding volumeThe base of a solid in the region bounded by the two parabolas       y    2    =  8  x and       x    2    =  8  y. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?The answer choices are:A)             288      π        35  B)             576      π        35  C)             144      π        35  D)   8  πI started off by writing the integral like this:      ∫          0              8            1    2    π            (            1    2              (            8    x    −            x      2        8              )                          )              2      d  xthen I simplified it to       π    8        ∫          0              8                  (            8    x    −            x      2        8                      )              2      d  xI can&#039;t think of anyway to solve this problem from here, without using a calculator.

slapadabassyc · Accepted Answer

Step 1Expand the binomial term:  V  =      π    8        ∫          x      =      0        8              (              8        x            −                        x          2                8            )        2      d  x  =      π    8        ∫          x      =      0        8    8  x  −  2      8    x    ⋅            x      2        8    +            x      4        64      d  x  .Step 2Now integrate term by term:  V  =      π    8              [      4              x        2            −                        2                7                    x                  7                      /                    2                    +                        x          5                          5          (          64          )                    ]              x      =      0        8    .