Certainly, let’s tackle this problem step-by-step. The goal is to find the area in the first quadrant below the line $y = 1$ and between the circles $x^2 + y^2 = 4$ and $x^2 - 2x + y^2 = 0$, using polar coordinates.

Step 1: Interpret the Regions Defined by the Curves

1. Circle 1: $x^2 + y^2 = 4$

   • This represents a circle centered at the origin with radius $2$.

2. Circle 2: $x^2 - 2x + y^2 = 0$

   • Completing the square for the $x$ terms: $(x - 1)^2 + y^2 = 1$
   • This represents a circle centered at $(1, 0)$ with radius $1$.

3. Line: $y = 1$

   • This is a horizontal line intersecting the first quadrant.

Step 2: Convert the Equations to Polar Coordinates

1. For the circle $x^2 + y^2 = 4$: $r^2 = 4 \Rightarrow r = 2$

2. For the circle $(x - 1)^2 + y^2 = 1$:

   • Convert $x = r\cos\theta$ and $y = r\sin\theta$: $(r\cos\theta - 1)^2 + (r\sin\theta)^2 = 1$
   • Expanding and simplifying: $r^2 - 2r\cos\theta + 1 = 1$ $r^2 = 2r\cos\theta \Rightarrow r = 2\cos\theta$

3. For the line $y = 1$: $r\sin\theta = 1 \Rightarrow r = \frac{1}{\sin\theta}$

Step 3: Set up the Integral

• We need to find the bounds for $r$ and $\theta$ where the area lies between $r = 2\cos\theta$ and $r = 2$, and below the line $r = \frac{1}{\sin\theta}$.

1. Angular Boundaries: Since we're in the first quadrant, $0 \leq \theta \leq \frac{\pi}{2}$.
2. Radial Boundaries: The radius $r$ will range from the inner circle $r = 2\cos\theta$ to the outer circle $r = 2$, but below the line $y = 1$, where $r = \frac{1}{\sin\theta}$.

Thus, the bounds on $r$ will be:

• $r = 2\cos\theta$ to $r = 2$ for $0 \leq \theta \leq \arcsin\left(\frac{1}{2}\right)$.
• For $\arcsin\left(\frac{1}{2}\right) \leq \theta \leq \frac{\pi}{2}$, the upper bound on $r$ is $r = \frac{1}{\sin\theta}$.

Step 4: Set Up and Evaluate the Integral

The area $A$ can be represented by: $A = \int_0^{\arcsin(1/2)} \int_{2\cos\theta}^{2} r \, dr \, d\theta + \int_{\arcsin(1/2)}^{\pi/2} \int_{2\cos\theta}^{\frac{1}{\sin\theta}} r \, dr \, d\theta$

1. Evaluate the first integral: $\int_0^{\arcsin(1/2)} \int_{2\cos\theta}^{2} r \, dr \, d\theta = \int_0^{\arcsin(1/2)} \left[ \frac{r^2}{2} \right]_{2\cos\theta}^{2} d\theta$

2. Evaluate the second integral: $\int_{\arcsin(1/2)}^{\pi/2} \int_{2\cos\theta}^{\frac{1}{\sin\theta}} r \, dr \, d\theta = \int_{\arcsin(1/2)}^{\pi/2} \left[ \frac{r^2}{2} \right]_{2\cos\theta}^{\frac{1}{\sin\theta}} d\theta$

You can now solve these integrals to find the area.

Would you like further assistance with the integration steps?

Follow-up Questions

1. How can polar coordinates simplify integration problems involving circular regions?
2. What are some common applications of polar coordinates in physics and engineering?
3. How do you convert other shapes, like ellipses, to polar coordinates?
4. What other coordinate systems can be used for evaluating areas and volumes?
5. How does the region of integration change if we move to different quadrants?

Tip:

Always sketch the region to better visualize the boundaries before setting up your integral in polar coordinates.