Chapter 2

Previous: Chapter 1

Let \(f\) be a continuous function on the rectangular domain \(\{(x,y): a \leq x \leq b , c \leq y \leq d\}\)

Integration

\(\int_{a}^{b} \int_{c}^{d} f(x,y) dydx = \int_{d}^{c} \int_{a}^{b}f(x,y)dxdy\)

Let f be defined on the rectangular domain \(\{(x,y): a \leq x \leq b , c \leq y \leq d\}\)

\(\int_{a}^{b} \int_{c}^{d} f(x,y) dydx = (\int_{d}^{c} a(x)dx)(\int_{d}^{c}b(y)dy)\)

Type I Domain

A region D is said to be a type I region if it is bounded below two curves, say \(y = g(x)\) and \(y=h(x)\), over an interval of the form \(a\leq x\leq b\), where on curve lies entirely above the other.

\(\int \int_{D}f(x,y)dA = \int_{a}^{b}\int^{h(x)}_{g(x)}f(x,y) dydx\)

Type II Domain

\(\int \int_{D}f(x,y)dA = \int_{c}^{d}\int^{h(y)}_{g(y)}f(x,y) dxdy\)

A region D is said to be a type II region if it is bounded between two curves, say \(x=g(y)\) and \(x=h(y)\), over an interval of the form \(a\leq x\leq b\) where one curve lies entirely to the right of the other.

If the domain is both Type I and Type II order can be swapped

Polar coordinates conversion

\(x = r cos \theta\)
\(y=rcos \theta\)
\(r^{2}=x^{2} + y^2\)

Let f be a cont. function defined on the polar rect.
\(D \{(r,\theta): g(\theta) \leq r \leq h(\theta)\) and \(\alpha \leq \theta \leq \beta\}\), where \(0 \leq \beta - \alpha \leq 2\pi\)\

\(\int \int_{D}f(x,y) = \int_\alpha^\beta\int_{g(\theta)}^{h(\theta)}r drd\theta\)

In case \(g(\theta) = 0\)

\(\int \int_{D}f(x,y) = \int_\alpha^\beta\int^{h(\theta)}_{0}r drd\theta\)

Next: Chapter 3