Section 4.2 Double Integrals over General Regions

4.2 Double Integrals over General Regions

In this section, we learn the double integral over general regions, not just rectangle region. In calculus I, we learn to integrate the area between two function $g_1(x)\le g_2(x)$ over the interval $[a,b]$

$${\int }_a^b{g}_2(x)-g_1(x)dx.$$

The idea is coming from that we cut the region bounded by $g_1(x)$, $g_2(x)$, $x=a$ and $x=b$ into subintervals $[x_{i-1},x_i]$. 

Now suppose $f(x,y)=1$ for all $x,y$ in the region and we want to find the volume of the solid bounded by $f(x,y)$ and $xy$-plane over a region $D=\{(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)\}$. The base of the solid is 

$${\int }_a^b{g}_2(x)-g_1(x)dx$$  and the height is $1$. Hence the volume is 

$${\int }_a^b{g}_2(x)-g_1(x)dx={\int }_a^b{\int }_{g_1(x)}^{g_2(x)}1dydx.$$

Can we do the similar approach for any $f(x,y)$ over $D$? 

Recall that we can do the double integration over $R=[a,b]\times [c,d]$ via integrate with respect to one variable first and the other later. The idea is that we cut a solid into thin area regions. For example, we can integrate $f(x,y)$ with respect to $y$ first to obtain a function with respect to $x$ only, $A(x)$, then we integrate with respect to $x$ later:

$${\int }_a^b{\int }_c^{d}f(x,y)dydx={\int }_a^{b}A(x)dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}A(x_i)\mathrm{△}x$$

Now for general region $D=\{(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)\}$, we cut the interval $[a,b]$ into subintervals $[x_{i-1},x_i]$ then $A(x_i)$ is changing depending on $f(x_i,y)$ and $g_1(x_i)\le y\le g_2(x_i)$. Hence $A(x_i)={\int }_{g_1(x_i)}^{g_2(x_i)}f(x_i,y)dy$. 

Theorem: Fubini’s Theorem (Strong Form)

Let $f(x,y)$ be a continuous function over $D=\{(x,y)|a\le x\le b,g_1(x)\le y\le g_2(x)\}$ (we call this type I region), then 

$${\iint }_{D}f(x,y)dA={\int }_a^b[{\int }_{g_1(x)}^{g_2(x)}f(x,y)dy]dx.$$

If $f(x,y)$ is a continuous function over $D=\{(x,y)|h_1(y)\le x\le h_2(y),c\le y\le d\}$ (we call this type II region), then 

$${\iint }_{D}f(x,y)dA={\int }_c^d[{\int }_{h_1(y)}^{h_2(y)}f(x,y)dx]dy.$$

Example 1: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=x{y}^2$ and $D=\{(x,y)|0\le x\le 1,x^2\le y\le x\}$ 

Exercise 1: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=xy$ and $D=\{(x,y)|2\le x\le 3,2x\le y\le 3x\}$ 

Example 2: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=y\text{sin}(x)$ and $D=\{(x,y)|2y\le x\le 3y,0\le y\le 1\}$ 

Exercise 2: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=e^{x}y$ and $D=\{(x,y)|0\le x\le 2y,2\le y\le 3\}$ 

Example 3: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=\frac{x^2}{y^2+3}$ and $D=\{(x,y)|0\le x\le \sqrt[3]{y},0\le y\le 1\}$.

Exercise 3: Compute the double integral ${\iint }_{D}f(x,y)dA$ where $f(x,y)=\frac{y}{x^2+1}$ and $D=\{(x,y)|0\le x\le 2,0\le y\le \sqrt{x}\}$.

Example 4: Compute the double integral $\int {\int }_{D}xydA$ where $D$ is the region bounded by $y=\sqrt{x}$ and $x=2y$.

Exercise 4: Compute the double integral $\int {\int }_{D}xydA$ where $D$ is the region bounded by $y=x^2$ and $x=y$.

Example 5: Compute the double integral $\int {\int }_D{x}^2{e}^{xy}dA$ where $D$ is the region bounded by $y=x$, $x=2$ and $y=0$.

Exercise 5: Compute the double integral $\int {\int }_D{y}^2{e}^{xy}dA$ where $D$ is the region bounded by $y=x$, $y=1$ and $x=0$.

Example 6: Sketch the solid whose volume is given by the iterated integral

$${\int }_0^1{\int }_0^{1-x}(1-x-y)dydx$$

Example 7: Find the volume of the solid bounded by $z=y$, $y=2x$, $x+y=6$ and $z=0$.

Group work

1. Sketch the solid whose volume is given by the iterated integral

$${\int }_0^1{\int }_0^y(1-y^2)dxdy.$$

2. Find the volume of the solid bounded by planes $z=x$, $x=0$, $y=x$, $x+y=4$ and $z=0$.

3. Sketch the solid whose volume is given by the iterated integral

$${\int }_0^1{\int }_0^{1-y^2}(1-y)dxdy.$$