(a) Transform the expression (x − a)^2 + y^2 = a^2 into polar coordinates.
(b) Sketch the region R bounded by the curve given in part (a).
(c) Use a double integral in polar coordinates to find the area of the region R.
You know it's a circle of radius a with center at (a,0)
x^2+y^2-2ax+a^2 = a^2
r^2 = 2arcosθ
r = 2a cosθ
Integrate, but realize that the circle is traced twice as θ goes from 0 to 2π. I hope you come up with πa^2 as the area!
(a) To transform the expression (x - a)^2 + y^2 = a^2 into polar coordinates, we can substitute x and y with their respective polar form expressions.
In polar coordinates, x = r*cos(theta), and y = r*sin(theta).
Substituting these expressions into the equation, we have:
(r*cos(theta) - a)^2 + (r*sin(theta))^2 = a^2
Expanding the equation, we get:
r^2*cos^2(theta) - 2*a*r*cos(theta) + a^2 + r^2*sin^2(theta) = a^2
Simplifying the expression, we have:
r^2*(cos^2(theta) + sin^2(theta)) - 2*a*r*cos(theta) = 0
Since cos^2(theta) + sin^2(theta) = 1, we can simplify further:
r^2 - 2*a*r*cos(theta) = 0
This is the equation in polar coordinates for the given expression.
(b) To sketch the region R bounded by the curve given in part (a), we need to understand the shape of the curve.
The equation in part (a): r^2 - 2*a*r*cos(theta) = 0
This equation represents a circle centered at (a, 0) with radius a, in the Cartesian coordinate system. In polar coordinates, the equation represents a circle with center at (a, pi) and radius a.
So, the region R is the area inside this circle.
(c) To find the area of the region R using a double integral in polar coordinates, we can use the formula:
Area = ∫∫R r dr dθ
Since we are dealing with a circle, we can specify the limits of integration. The radius of the circle is a, and the angle θ ranges from 0 to 2π to cover the whole circle.
Thus, the limits of integration for the double integral are:
r: 0 to a
θ: 0 to 2π
The integral becomes:
Area = ∫[0 to 2π] ∫[0 to a] r dr dθ
Evaluating the integral, we get:
Area = ∫[0 to 2π] (1/2)*a^2 dθ
Integrating with respect to θ, we have:
Area = (1/2)*a^2 * θ |[0 to 2π]
Applying the limits, we get:
Area = (1/2)*a^2 * (2π - 0) = π*a^2
Thus, the area of the region R is π times the square of the radius a.