Multivariable calculus and vector analysis

Area of region example

Find area of region bounded by parabola x = y² and the line y = x.

Graph:

We’ll let y go from 0 to 1. Then x goes from y² to y.

Area is

∫ 01 ∫ y2ydxdy	= ∫ 01dy
	= ∫ 01(y - y2)dy
	= -01
	= - =

Note: if asked to integrate a function f(x,y) over that region, you’d need to compute

∫ 01 ∫ y2yf(x,y)dxdy.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

3a. Double integrals

• Double integrals
• Double integrals as iterated integrals
• Double integral examples
• Double integrals as volume or area
• Change order of integration examples
• Area of region example
• Changing order of integration is not always feasible

3b. Triple Integrals

3c. Changing variables

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material