The Area Under the Witch of Agnesi

The area under the witch of Agnesi is discussed. Real-variable proofs of typically appeal to the fact that the inverse tangent is an anti-derivative for the integrand or that the differential equation tan = 1 + tan2 holds on the open interval. Complex-variable proofs typically appeal to Cauchy's residue theorem. The proof below requires neither Cauchy's theorem nor knowledge of any transcendental functions, and also connects the value of the integral more directly with the geometrical definition of pi.