3.2 Double Integrals in Polar Coordinates

The left hand sketch below contains the points, $(x_1,y_1)$, $(x_3,y_3)$, $(x_5,y_5)$, that are on the axes. The right hand sketch below contains the points, $(x_2,y_2)$, $(x_4,y_4)$, that are not on the axes.

$r_1 = 3$, $\theta_1=0$ $r_2 = \sqrt{2}$, $\theta_2=\frac{\pi}{4}$ $r_3 = 1$, $\theta_3=\frac{\pi}{2}$ $r_4 = \sqrt{2}$, $\theta_4=\frac{3\pi}{4}$
$r_5 = 2$, $\theta_5=\pi$

$r$ is allowed to be negative.

(a) $\big(r_1=2\,,\,\theta_1= n\pi,\ n\text{ odd integer }\big)$ or $\big(r_1=-2\,,\,\theta_1= n\pi,\ n\text{ even integer }\big)$. In particular, $\big(r_1=-2\,,\,\theta_1= 0\big)$ has $r_1<0$ and $0\le\theta_1<2\pi$.

(b) $\big(r_2=\sqrt{2}\,,\,\theta_2= \nicefrac{\pi}{4} + 2n\pi\big)$ or $\big(r_2=-\sqrt{2}\,,\,\theta_2= \nicefrac{5\pi}{4} + 2n\pi\big)$, with $n$ integer. In particular, $\big(r_2=-\sqrt{2}\,,\, \theta_2= \nicefrac{5\pi}{4}\big)$ has $r_2<0$ and $0\le\theta_2<2\pi$.

(c) $\big(r_3=\sqrt{2}\,,\,\theta_3= \nicefrac{5\pi}{4} + 2n\pi\big)$ or $\big(r_3=-\sqrt{2}\,,\,\theta_3= \nicefrac{\pi}{4} + 2n\pi\big)$, with $n$ integer. In particular, $\big(r_3=-\sqrt{2}\,,\, \theta_3= \nicefrac{\pi}{4}\big)$ has $r_3<0$ and $0\le\theta_3<2\pi$.

(d) $\big(r_4=3\,,\,\theta_4= 0 + 2n\pi\big)$ or $\big(r_4=-3\,,\,\theta_4= \pi + 2n\pi\big)$, with $n$ integer. In particular, $\big(r_4=-3\,,\, \theta_4= \pi\big)$ has $r_4<0$ and $0\le\theta_4<2\pi$.

(e) $\big(r_5=1\,,\,\theta_5= \nicefrac{\pi}{2} + 2n\pi\big)$ or $\big(r_5=-1\,,\,\theta_5= \nicefrac{3\pi}{2} + 2n\pi\big)$, with $n$ integer. In particular, $\big(r_5=-1\,,\, \theta_5= \nicefrac{3\pi}{2}\big)$ has $r_5<0$ and $0\le\theta_5<2\pi$.

Compute, for each angle $\theta$, the dot product $\he_r(\theta)\cdot\he_\theta(\theta)$.

(a) Both $\he_r(\theta)$ and $\he_\theta(\theta)$ have length 1. The angle between them is $\frac{\pi}{2}$. The cross product is $\he_r(\theta) \times \he_\theta(\theta)=\hk$.

(b) Here is a sketch of $(x_i,y_i)$, $\he_r(\theta_i)$, $\he_\theta(\theta_i)$ for $i =1,3,5$ (the points on the axes)

and here is a sketch (to a different scale) of $(x_i,y_i)$, $\he_r(\theta_i)$, $\he_\theta(\theta_i)$ for $i =2,4$ (the points off the axes).

Sketch $\llt a, b\rgt$ and $\llt A, B\rgt$. The trigonometric addition formulas

will help.

(a) $a=r\cos\theta$, $b=r\sin\theta$

(b) $A=a\cos\varphi-b\sin\varphi$, $B=b\cos\varphi+a\sin\varphi$

(a) $\displaystyle\dblInt_\cR f(x,y)\,\dee{x}\,\dee{y} =\int_0^{\nicefrac{\pi}{4}}\dee{\theta} \int_0^2\dee{r}\ r\ f(r\cos\theta,r\sin\theta) =\int_0^2\dee{r} \int_0^{\nicefrac{\pi}{4}}\dee{\theta} \ r\ f(r\cos\theta,r\sin\theta)$

(b) $\displaystyle\dblInt_\cR f(x,y)\,\dee{x}\,\dee{y} =\int_0^{\nicefrac{\pi}{2}}\dee{\theta} \int_1^2\dee{r}\ r\ f(r\cos\theta,r\sin\theta) =\int_1^2\dee{r} \int_0^{\nicefrac{\pi}{2}}\dee{\theta} \ r\ f(r\cos\theta,r\sin\theta)$

(c) $\displaystyle \dblInt_\cR f(x,y)\,\dee{x}\,\dee{y} =\int_0^{\nicefrac{\pi}{2}}\dee{\theta} \int_0^{2\cos\theta}\dee{r}\ r\ f(r\cos\theta,r\sin\theta) =\int_0^2\dee{r} \int_0^{\arccos\frac{r}{2}}\dee{\theta} \ r\ f(r\cos\theta,r\sin\theta)$

(d)

(a)

(b)

(c)

(a) $\frac{a^3}{6}\big[\sqrt{3}+1\big]$ (b) $\frac{2}{3}$ (c) $\frac{1}{3}$ (d) $-\pi$

$\pi \left[\frac{4}{3}\sqrt{2}-\frac{7}{6}\right] \approx 2.26$

$\frac{64}{9}a^3$

$\frac{128}{15}a^3$

(a)

(b) $\frac{32}{9}$

$\frac{16\pi}{5}$

(a)

(b) $M = \int_0^{\pi/4}\dee{\theta}\int_{\sqrt{2}}^2\dee{r}\ r\, \rho(r\cos\theta\,,\,r\sin\theta)$ (c) $\frac{1}{2}$

$\pi$

$1-\frac{1}{\sqrt{2}}$

(a)

(b) $\frac{\pi}{12}\big[5\ln(5)-4\big]$

$\frac{\sqrt{3}}{2}-\frac{\pi}{6}$

(a) $D = \Set{(r\cos\theta\,,\,r\sin\theta)} {-\nicefrac{\pi}{2}\le\theta\le\nicefrac{\pi}{4},\ 0\le r\le 2\cos\theta}$

(b) $\text{Volume}=\frac{40}{18\sqrt{2}} +\frac{16}{9}$

$2\frac{\pi+44/9}{\pi+8}\approx 1.442$

(a)

(b)

(c)

(a)

(b) $J = \int_0^{\sqrt{2}}\int_0^x \frac{y}{x} e^{x^2+y^2}\ \dee{y}\,\dee{x} + \int_{\sqrt{2}}^2\int_0^{\sqrt{4-x^2}} \frac{y}{x} e^{x^2+y^2}\ \dee{y}\,\dee{x}$

(c) $\frac{1}{4}\left[e^4-1\right]\ln 2$

$\frac{\pi}{8}ab$

About 3.5” above the bottom

(a) $\pi\big(e^9-1\big)\approx 25,453$

(b) $\int_0^1 \dee{x}\int_x^{2-x}\dee{y}\ e^{x^2+y^2}$