How to Find the Volume of a Spherical Cap Using Calculus

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The general approach of calculus in calculating volumes of objects with curved surfaces is based on the main theory of integration. In essence, one chops up the three dimensional object into smaller and smaller slices, and approximates the volume of each of those slices using a simpler shape. To find the volume of a spherical cap, the easiest formulation is to imagine a pile of wide, short cylinders on top of one another. The volume is calculated as the height of each of these cylinders goes to zero, generating finer and finer approximations.

Things You'll Need

  • pencil
  • paper
  • calculator (optional)

Writing down the integral

  • Determine the diameter or radius of your spherical cap at its widest part.

  • Determine the height of the spherical cap.

  • Square the numbers from Steps 1 and 2, and add them together. Divide this number by twice the number you found in step 2. This gives you R, the radius of the sphere that the spherical cap was cut from.

  • Write down "V = ", followed by the integration symbol.

  • Subtract the number you found in Step 2 from R, and write this number at the bottom of the integration symbol.

  • Write down the value of R at the top of the integration symbol.

  • Write down pi, followed by a parenthesis, after the integration symbol.

  • Square the value of R, and write that after the parenthesis, followed by a minus sign.

  • Write down "x^2", followed by the closing parenthesis. Finish writing out the integral with "dx."

Evaluating the integral

  • Multiply pi into the parenthesis, yielding pi*x^2 subtracted from a constant.

  • Evaluate the first term of the integral by multiplying the constant by the height of the spherical cap (really, R - a, the two endpoints of the integral), and moving it outside the integral. The equation should now be of the form "V = C(R - a) -- [definite integral from a to R] pi*x^2 dx", where C is the square of R times pi, and a is R minus the height of the spherical cap.

  • The remaining integral evaluates to 1/3pi(R^3) -- 1/3pi(a^3). Thus the overall formula for the volume of a spherical cap is V = C(R - a) -- 1/3pi(R^3) + 1/3pi(a^3), where C and a are as described in Step 2, and R is as described in Step 3 of the previous section.

  • Substituting R minus the height of the spherical cap ("h") for a, evaluating the cubes, and simplifying yields V = 1/3pih^2*(3R -- h), the standard algebraic formula for the volume of a spherical cap.

References

  • Photo Credit Hemera Technologies/AbleStock.com/Getty Images
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