Double Integration: Polar form

Double Integration in Polar Coordinates

Animated Step-by-Step Solutions

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Example 1: Area of Circle $x^2+y^2 \le 4$

Evaluate: $$\iint_R 1 \, dA$$

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Step 1: Convert to Polar Coordinates

$x = r\cos\theta$, $y = r\sin\theta$

$x^2+y^2 = r^2$

$dA = r\,dr\,d\theta$

Step 2: Limits

$0 \le r \le 2$, $0 \le \theta \le 2\pi$

Step 3: Integral

$$ \int_0^{2\pi}\int_0^2 r\,dr\,d\theta $$

$$ \int_0^2 r\,dr = \frac{r^2}{2}\Big|_0^2 = 2 $$

$$ \int_0^{2\pi} 2\,d\theta = 4\pi $$

Final Answer: $4\pi$

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Example 2: Quarter Circle Integral

Evaluate: $$\iint_R (x^2+y^2)\, dA$$

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Step 1: $x^2+y^2 = r^2$

Limits:

$0 \le r \le 3$

$0 \le \theta \le \frac{\pi}{2}$

$$ \int_0^{\pi/2}\int_0^3 r^3\,dr\,d\theta $$

$$ \int_0^3 r^3 dr = \frac{81}{4} $$

$$ \frac{81}{4} \cdot \frac{\pi}{2} = \frac{81\pi}{8} $$

Final Answer: $\frac{81\pi}{8}$

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Example 3: Annular Region

Evaluate: $$\iint_R 1\,dA$$

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Region: $1 \le r \le 2$, $0 \le \theta \le 2\pi$

$$ \int_0^{2\pi}\int_1^2 r\,dr\,d\theta $$

$$ \int_1^2 r dr = \frac{3}{2} $$

$$ \frac{3}{2}\cdot 2\pi = 3\pi $$

Final Answer: $3\pi$

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Example 4: Cardioid $r=1+\cos\theta$

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$$ \int_0^{2\pi}\int_0^{1+\cos\theta} r^2\,dr\,d\theta $$

$$ \int_0^{1+\cos\theta} r^2 dr = \frac{(1+\cos\theta)^3}{3} $$

Expand and integrate term-wise over $0$ to $2\pi$.

Final Answer: $\frac{5\pi}{2}$

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Example 5: Gaussian Integral

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$$ \iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dA $$

$$ = \int_0^{2\pi}\int_0^\infty e^{-r^2} r\,dr\,d\theta $$

Let $u=r^2$, $du=2rdr$

$$ \int_0^\infty e^{-r^2}r\,dr = \frac{1}{2} $$

$$ \frac{1}{2}\cdot 2\pi = \pi $$

Final Answer: $\pi$

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