Double integral in polar coordinates - concentric circles

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Question statement

Compute the integral

$\iint_R \frac{y^2}{\sqrt{x^2 + y^2}}\, dx\, dy$

where $R$ is the region between the circles $x^2 + y^2 = \var{r1}$ and $x^2 + y^2 = \var{r2}$ .

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			Name	Type	Generated Value

r1	integer	2
r2	integer	4
ans	number	58.643062867
diagram	jsxgraphboard	HTML node

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Generated value: `integer`

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"Part b)" → "Gap 2." - Correct answer
"Part c)" - Correct answer

Advice

Recall that ew pass to polar coordinates using $x=r\cos(\theta)$ and $r\sin(\theta)$.

a) Plug the values above into $ \frac{y^2}{\sqrt{x^2 + y^2}}$ we get

\[ \frac{r^2\sin^2(\theta)}{\sqrt{r^2\cos^2(\theta) + r^2\sin^2(\theta)}} = \frac{r^2\sin^2(\theta)}{r\sqrt{\cos^2(\theta) + \sin^2(\theta)}} = r\sin^2(\theta).\]

b) The area we are interested is the area between the concentric circles below (this shape is called and annulus)

{diagram}

This are can be described by the inequalities

\[0<\theta\leq 2\pi \quad \mbox{ and }\quad \var{r1}<r<\var{r2}\]

in polar coordinates.

c) Note that the order the integration does not matter as boundaries for both variables are constants. The integral thus can be written as

\[\int_{\var{r1}}^{\var{r2}}\int_{0}^{2\pi} r^2\sin^2(\theta)\, d\theta\,dr.\]

\[\int_{\var{r1}}^{\var{r2}}\int_{0}^{2\pi} r^2\sin^2(\theta)\, d\theta\,dr = \int_{\var{r1}}^{\var{r2}}r^2 \int_{0}^{2\pi} \sin^2(\theta)\, d\theta\,dr = -\int_{\var{r1}}^{\var{r2}}r^2 \left[\frac{\sin(2\theta) - 2\theta}{4}\right]_0^{2\pi}\,dr
= \\[3mm ]\int_{\var{r1}}^{\var{r2}}r^2 \pi\,dr = \pi\left[\frac{r^3}{3}\right]_{\var{r1}}^{\var{r2}} = \pi\left(\frac{\var{r2}^3 - \var{r1}^3}{3}\right) = \var{ans}.\]

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Double Integrals by Ugur Efem in Ugur's workspace.

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Double integral in polar coordinates - concentric circles

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Double integral in polar coordinates - concentric circles

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ugur Efem 2 years ago

Ugur Efem 2 years ago

Ugur Efem 2 years, 1 month ago

Error in variable testing condition

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Generated value: integer

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Mathematical expression

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Checking accuracy

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Generated value: `integer`