Set up the polar integral that represents the area of the region R in the xy-plane enclosed by the circle x² + y² = 4, above the line y = 1, and below the line y = √3x.
Understand the Problem
The question is asking to set up a polar integral for the area of a specific region in the xy-plane. The area is bound by a circle and two lines, so we need to express the given boundaries in polar coordinates and establish the integral limits accordingly.
Answer
$$ A = \int_{\theta=0}^{\frac{\pi}{3}} \int_{r=\frac{1}{\sin(\theta)}}^{2} r \, dr \, d\theta $$
Answer for screen readers
The polar integral representing the area is
$$ A = \int_{\theta=0}^{\frac{\pi}{3}} \int_{r=\frac{1}{\sin(\theta)}}^{2} r , dr , d\theta $$
Steps to Solve
- Identify the boundaries of the region
The area is bounded by the circle $x^2 + y^2 = 4$, which corresponds to a radius of 2 centered at the origin. The lines are $y = 1$ and $y = \sqrt{3}x$.
- Convert boundaries to polar coordinates
In polar coordinates:
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The circle becomes: $$ r = 2 $$
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The line $y = 1$ converts to: $$ r \sin(\theta) = 1 \quad \Rightarrow \quad r = \frac{1}{\sin(\theta)} $$
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The line $y = \sqrt{3}x$ converts to: $$ r \sin(\theta) = \sqrt{3}r \cos(\theta) \quad \Rightarrow \quad \tan(\theta) = \sqrt{3} \quad \Rightarrow \quad \theta = \frac{\pi}{3} $$
- Determine the limits of integration
The region in the circle is limited by the lines $y = 1$ and $y = \sqrt{3}x$.
- For the line $r = \frac{1}{\sin(\theta)}$, the limits of $\theta$ are from $0$ to $\frac{\pi}{3}$.
- Set up the polar integral
The area $A$ can be expressed as: $$ A = \int_{\theta=0}^{\frac{\pi}{3}} \int_{r=1/\sin(\theta)}^{2} r , dr , d\theta $$
This integral describes the area of region $R$ bounded by the specified curves.
The polar integral representing the area is
$$ A = \int_{\theta=0}^{\frac{\pi}{3}} \int_{r=\frac{1}{\sin(\theta)}}^{2} r , dr , d\theta $$
More Information
This setup is essential for calculating areas in polar coordinates, particularly in instances involving circles and straight lines. The transformation helps to easily integrate across complex shapes.
Tips
- Forgetting to convert all boundaries to polar coordinates correctly.
- Confusing the limits of integration, especially when determining where the lines intersect the polar boundaries.
- Not considering the area correctly when the curves enclose more than one section.
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