Double integrals extend single-variable integration to two dimensions, allowing us to calculate volumes under surfaces over rectangular regions. We use iterated integration, applying Fubini's Theorem to switch integration order when helpful.

These integrals are powerful tools for finding volumes and average values over rectangular regions. By choosing the right order of integration, we can simplify complex calculations and solve problems more efficiently.

Understanding Double Integrals over Rectangles

Evaluation of double integrals

• Double integrals extend single variable integration to two variables representing volume under surface over rectangular region
• Notation $\int\int_R f(x,y) dA$ where R denotes rectangular region
• Limits of integration outer corresponds to one variable inner to other
• Iterated integration process integrates wrt one variable treating other as constant then integrates result wrt second variable
• Fubini's Theorem allows changing integration order facilitating simpler calculations (x-y to y-x)

Volume calculation with double integrals

• Double integral geometrically represents volume under surface $z = f(x,y)$ over region R
• Volume calculation setup $V = \int\int_R f(x,y) dA$
• Identify function $f(x,y)$ representing surface (paraboloid, plane)
• Determine rectangular region R boundaries (x from 0 to 2, y from 1 to 3)
• Evaluate integral to find volume using iterated integration

Average value over rectangular regions

• Average value in two dimensions extends 1D concept to surfaces
• Formula $f_{avg} = \frac{1}{A} \int\int_R f(x,y) dA$
• Calculate rectangular region area $A = (b-a)(d-c)$ for rectangle $[a,b] \times [c,d]$
• Set up and evaluate double integral of function over region
• Divide integral result by region area to obtain average value

Order of integration in rectangles

• Rectangular regions allow flexible integration order simplifying calculations
• Identify outer and inner integrals outer determines first variable to integrate wrt
• Consider function complexity wrt each variable choose order simplifying process
• Fubini's Theorem justifies changing integration order $\int_a^b \int_c^d f(x,y) dy dx = \int_c^d \int_a^b f(x,y) dx dy$
• Analyze function structure to determine optimal integration order (polynomial, trigonometric)