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Double integral over a rectangle
Example
Let $D$ be the rectangle $0 \leq x \leq 3$, $0 \leq y \leq 2$. Use a Riemann sum corresponding to the partition of $D$ into six unit squares with points selected at the centre of each square to approximate $$\iint_D (x^2 + y)\,dA$$
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Double integral over a general domain
Theorem:
Integrability on bounded domains
If $f$ is continuous on a closed, bounded domain $D$ whose boundary consists of finitely many curves of finite length, then $f$ is integrable on $D$.
Regular domains
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Theorem:
Iteration of double integrals
If $f(x,y)$ is continuous on the bounded y-simple domain $D$ given by $a \leq x \leq b$ and $c(x) \leq y \leq d(x)$, then $$\iint_D f(x,y)\,dA = \int_a^b dx \int_{c(x)}^{d(x)} f(x,y)\,dy$$ Similarly, if $f$ is continuous on the bounded x-simple domain $D$ given by $c \leq y \leq d$ and $a(y) \leq x \leq b(y)$, then $$\iint_D f(x,y)\,dA = \int_c^d dy \int_{a(y)}^{b(y)} f(x,y)\,dx$$
Notation
The symbol $dA$ in the double integral is replaced by the $dx$ and the $dy$ in the iterated integrals. Accordingly, $dA$ is frequently written $dx\,dy$ or $dy\,dx$. The three expressions $$\iint_D f(x,y)\,dx\,dy, \qquad \iint_D f(x,y)\,dy\,dx, \qquad \iint_D f(x,y)\,dA$$ all stand for the double integral of $f$ over $D$. The order of $dx$ and $dy$ becomes important only when the double integral is iterated.
Example
Find the volume of the solid lying above the rectangle $Q = \{(x,y) \mid 0 \leq x \leq 1,\; 1 \leq y \leq 2\}$ and below the plane $z = 4 - x - y$.
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Example
Evaluate $\displaystyle\iint_T xy\,dA$ where $T$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(1,1)$.
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Example
Evaluate the iterated integral $$I = \int_0^1 dx \int_{\sqrt{x}}^{1} e^{y^3}\,dy$$
Theorem:
Change to polar coordinates
If $f(x,y)$ is continuous on a domain $D$ in the $xy$-plane, and $D$ is described in polar coordinates by $\alpha \leq \theta \leq \beta$ and $a(\theta) \leq r \leq b(\theta)$, then $$\iint_D f(x,y)\,dA = \int_{\alpha}^{\beta} d\theta \int_{a(\theta)}^{b(\theta)} f(r\cos\theta,\, r\sin\theta)\;r\,dr$$
Example
Find the volume of the solid under the paraboloid $z = 1 - x^2 - y^2$ and above the $xy$-plane.
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Example
Find the volume of the solid lying in the first octant, inside the cylinder $x^2 + y^2 = a^2$, and under the plane $z = y$.
The Gaussian integral
Show that $$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$