A GEOMETRIC APPROACH TO A TRIGONOMETRIC INTEGRAL

Russell Euler Northwest Missouri State University

An example given to a second semester calculus class consisted of evaluating

∫_{0}^{π} sin^{2} x dx .

The standard ploy used to determine this integral is to first change the integrand by using the trigonometric identity

(1)  sin^{2} x = \frac{1}{2}(1 − cos 2x) .

After carrying out the details to obtain the answer, a student with an apparent phobia for trigonometry inquired as to whether the solution could be obtained without using the half-angle formula (1). The following alternate method yielded an acceptable approach.

By considering the areas of the regions shaded in figures 1 and 2, it is clear that

(2)  ∫_{0}^{π} sin^{2} x dx = ∫_{0}^{π} cos^{2} x dx .

Substituting (2) into the left-hand side of the identity

∫_{0}^{π}(sin^{2} x + cos^{2}x) dx = ∫_{0}^{π} 1 dx = π

gives

2 ∫_{0}^{π} sin^{2} x dx = π ,

and hence the desired result after dividing by 2.