Find the surface area of a sphere by setting up a double integral for the top hemisphere and evaluating it using polar coordinates.

Vault Of Euclid

Surface Area of a Sphere via Double Integration

Multiple integrals core

The surface area of a sphere. Find the surface area of the sphere with radius $a$ centered at the origin, whose top hemisphere has equation $f(x, y) = \sqrt{a^2 - x^2 - y^2}$.

The solution and a full step-by-step explanation are here. Sign in to read them.

What you need first

Partial derivatives
Double integrals
Polar coordinates
Integration by substitution

What this problem tests

Tests the ability to set up a surface area integral, convert it to polar coordinates, and evaluate the resulting integral using substitution.

This is a classic application of double integrals in a multivariable calculus course, demonstrating how to compute surface area over a circular region.

Surface Area of a Sphere via Double Integration

What you need first

What this problem tests

More on Multiple integrals