The base of a solid

Question

The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x² Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Answer 1

Problems like this are difficult without a drawing.   However, a few things can be noted from the text alone.

 

First:  the cross sections are perpendicular to the x axis, therefore the integration will be over x.

 

 Second, the curves intersect at two points  (0,0)  and (1,1), therefore the range of integration will be from 0 to 1.

 

  Third, the cross sections are semicircles.   The area of a semicircle in terms of its diameter is (pi/8) d². 

 

  So the integral for the volume is  

      V =  ∫ (pi/8) d² dx.    

  With a aid of a rough sketch, it can be seen that d = x -x²  for this problem.  So

 

  V =  ∫  (pi/8) [ x - x²]²  dx        { limits:  from 0 to 1  }

 

  V =  (pi/8) [ x² - 2 x³ + x⁴ ]  dx     {limigts :  from 0 to 1}  {integrate term by term using power law}

  

 V =  (pi/8) [ 1/3 - 2/4 + 1/5]      {notice there is no contribution from the lower limit}

 

  V = pi/240