Find the area of the region bounded by: r=7-1sin(theta)
To find the area of the region bounded by the polar curve r = 7 - sin(theta), you can use the concept of integration.
First, let's plot the curve to get an idea of the region.
The polar equation r = 7 - sin(theta) represents a cardioid curve. It has a single loop and is symmetric about the x-axis. The curve reaches a maximum value of r = 8 and a minimum value of r = 6.
To find the area, we can break down the region into small sectors. The area of each sector can be approximated by a thin rectangle, and by summing up all the rectangles, we can calculate the total area.
The formula to calculate the area for a thin rectangle in polar coordinates is A = (1/2) * r^2 * d(theta), where r is the radius and d(theta) is the small change in the angle theta.
Now, we need to set up the integral to find the area. The limits of integration for theta will depend on the region you want to find the area for. Let's assume we want to calculate the area for the full loop of the curve, which is from theta = 0 to theta = 2*pi.
The integral for the area can be written as:
A = ∫[0 to 2*pi] (1/2) * (7 - sin(theta))^2 * d(theta)
To evaluate this integral, you can expand the expression (7 - sin(theta))^2 and simplify the equation. Then, integrate the resulting expression with respect to theta from 0 to 2*pi.
Once you calculate the definite integral, you will have the area of the region bounded by the curve r = 7 - sin(theta).
Perform an area integration in polar coordinates.
V = (1/2)Integral[r(theta)]^2 d theta
(0 to 2 pi)