Problem: Find the surface of a torus with radius of a cross-section equal to r and the radius of rotation of the center of cross-section equal to R.

Solution: We start with a picture.

This torus is obtained by revolving a circle of radius r with center (R,0) about the y-axis. By symmetry, it is enough to calculate the surface of the upper half and then multiply it by 2.

The situation exactly fits the setting for which we have a formula and we know the function describing the upper half-circle (it satisfies the equation (x − R)² + y² = r²), so we can calculate (using methods from the "root of quadratics" box):

Note that the integral after the substitution z = x − R corresponds to the following picture:

In this setting one can express the whole circle using parametric equations x = rcos(t), y = rsin(t), for t from 0 to 2π. Thus we can calculate the surface using the appropriate formula.

Next problem
Back to Solved Problems - Applications