Double Integrals
Introduction
With single variable functions, an integral represents the area under a curve. With multi-variable equations, a double integral represents the volume under a surface.
The Essentials
The volume under a surface is described by the double integral
Just like \( dx \) is a small part of a curve integrated over in single variable calculus, \( dA \) is a small part of the area integrated over in multi-variable calculus. In rectangular coordinates with constant bounds \( \{(x,y)|a\leq x\leq b, c\leq y\leq d\} \) a double integral can be expressed as an iterated integral:
If the function being integrated in a double integral, \( f(x, y) \), can be split into a function of \( x \) and a function of \( y \) multiplied together \( f(x,y)=g(x)h(y) \) then the integral can be split into two integral multiplied together:
To integrate over non-constant bounds, put the functions in the bounds of the integral on the inside. To integrate in polar coordinates, the function and bounds are converted to polar coordinates and \( dA \) is changed to \( rdrd\theta \) instead of \( dxdy \).
Example
Evaluate the double integral of the function \( f(x,y)=x+y \) over the triangle made by the points (0,2), (1,0), (0,0).
We can set up this integral two different ways:
Solving the integral the second way gives us:
Practice
Evaluate the double integral of the function \( f(x,y)=x^2+y^2 \) over the region that is the circle with radius 5 centered on the origin minus the circle with radius 2 centered on the origin
Solution:
\( \frac{609\pi}{2} \)