Question

Вычислить площадь области ( с помощью перехода к полярной системе координат)

Answer 1

$x^2+y^2=e^2\; \; \to \; \; r^2cos^2\phi +r^2sin^2\phi=e^2,\\\\r^2(cos^2\phi +cos^2\phi )=e^2,\; \; r^2=e^2,\; r=e>0\\\\x^2+y^2=e^4\; \; \to \; \; r^2=e^4,\; \; r=e^2>0\\\\S=\iint r\, dr\, d\phi=4\int_0^{\frac{\pi}{2}}d\phi\int _{e}^{e^2}r\, dr=4\int_0^{\frac{\pi}{2}}(\frac{r^2}{2}}|_e^{e^2})d\phi =\\\\=4\cdot \frac{1}{2}\int _0^{\frac{\pi}{2}}(e^4-e^2)d\phi =2(e^4-e^2)\cdot \phi |_0^{\frac{\pi}{2}}=\\\\=2(e^4-e^2)(\frac{\pi}{2}-0)=\pi e^2(e^2-1)$

Заданная область -  кольцо между окружностями с центром в (0,0) и

радиусами $r=e,r=e^2.$

Answer 2

Calculating the Area of a Region Using Polar Coordinates

To calculate the area of a region using polar coordinates, we need to express the region in terms of polar equations and then integrate over the region.

Here are the steps to follow:

1. Express the region in polar coordinates: Convert the given region from Cartesian coordinates to polar coordinates. This involves expressing the boundaries of the region in terms of polar equations.

2. Determine the limits of integration: Find the limits of integration for the polar angle and the radial distance. The limits of integration for the polar angle will depend on the boundaries of the region in polar coordinates.

3. Set up the double integral: Set up the double integral in polar coordinates to calculate the area of the region. The integrand will be equal to 1, as we are calculating the area.

4. Evaluate the integral: Evaluate the double integral using the determined limits of integration.

Let's apply these steps to an example to illustrate the process.

Example: Find the area of the region bounded by the curves r = 2 - 2cos(theta) and r = 2.

Solution:

1. Express the region in polar coordinates: The region is bounded by the curves r = 2 - 2cos(theta) and r = 2. These equations represent two polar curves. The inner curve r = 2 - 2cos(theta) represents the boundary of the region.

2. Determine the limits of integration: The limits of integration for the polar angle theta will depend on the boundaries of the region. In this case, the region is bounded by the curves from theta = 0 to theta = 2pi.

3. Set up the double integral: The double integral to calculate the area of the region is given by:

A = ∫∫ r dr d(theta)

The integrand is equal to 1, as we are calculating the area.

4. Evaluate the integral: Integrating over the region, we have:

A = ∫[0 to 2pi] ∫[2 - 2cos(theta) to 2] r dr d(theta)

Evaluating this integral will give us the area of the region.

Please note that the specific limits of integration and the equations for the region will vary depending on the given problem. Make sure to adapt the steps accordingly.

I hope this explanation helps! Let me know if you have any further questions.