Finding volume of a sphereI am stuck on the following problem:A circle x 2 + y 2 = a 2 is rotated around the y-axis to form a solid sphere of radius a. A plane perpendicular to the y-axis at y = a 2 cuts off a spherical cap from the sphere. What fraction of the total volume of the sphere is contained in the cap?So far I have figured out the following:Rotating the cap on the y axis we get a height h starting from y = a 2 . The interval from y = 0 to y = a 2 (the region below the cap) should be: a − hI also know that the radius of the sliced disk, x, can be derived from the equation of the circle: x = a 2 − y 2 Since the area of a circle is A = π r 2 the area with respect to y for the circle should be: A ( y ) = π ( a 2 − y 2 )So to find the volume, we need to integrate the function: V = ∫ a 2 a π ( a 2 − y 2 ) d yI know where I should go, but I am not sure what to do about the constraint y = a 2 at this point. Should I integrate the terms with respect to y first and then plug in the value which is equal to y? Or should this be done before integrating?

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Finding volume of a sphereI am stuck on the following problem:A circle       x    2    +      y    2    =      a    2   is rotated around the y-axis to form a solid sphere of radius a. A plane perpendicular to the y-axis at   y  =      a    2   cuts off a spherical cap from the sphere. What fraction of the total volume of the sphere is contained in the cap?So far I have figured out the following:Rotating the cap on the y axis we get a height h starting from   y  =      a    2  . The interval from   y  =  0 to   y  =      a    2   (the region below the cap) should be:  a  −  hI also know that the radius of the sliced disk, x, can be derived from the equation of the circle:  x  =            a      2        −          y      2      Since the area of a circle is   A  =  π      r    2   the area with respect to y for the circle should be:  A  (  y  )  =  π  (      a    2    −      y    2    )So to find the volume, we need to integrate the function:  V  =      ∫          a      2        a    π  (      a    2    −      y    2    )  d  yI know where I should go, but I am not sure what to do about the constraint   y  =      a    2   at this point. Should I integrate the terms with respect to y first and then plug in the value which is equal to y? Or should this be done before integrating?

Sandra Randall · Accepted Answer

Step 1  y  =  a      /    2 is not a constraint; it&#039;s one of the integration bounds, and you&#039;ve already written it into the integral as an integration bound correctly. Now all you have to do is evaluate the integral.Step 2By the way, your derivation of the bounds seems unnecessarily complicated. There&#039;s no reason to invoke   y  =  0,  which doesn&#039;t play any special role here. The truncated sphere lies between -a and a/2 and the cap that&#039;s cut off lies between a/2 and a, so you integrate from a/2 to a; nothing to be subtracted or calculated there.

A circle x^{2}+y^{2}=a^{2} is rotated around the y-axis to form a solid sphere of radius a.

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