Question

Calculate the area of the circle r = 6 sin ? as an integral in polar coordinates. be careful to choose the correct limits of integration

Answer 1

To calculate the area of the circle r = 6sinθ in polar coordinates, we express the area element dA in terms of polar coordinates and find the limits of integration. The integral expression for the area is A = ∫0^π∫0^(6sinθ) (6sinθ)·dr·dθ.

Step-by-step explanation:

We can calculate the area of the circle with radius r = 6sinθ in polar coordinates as an integral. First, we need to express the area element dA in terms of polar coordinates. Using the formula dA = r·dr·dθ, we have dA = (6sinθ)·dr·dθ.

Next, we need to find the limits of integration. Since the wire is symmetrical about point O, we can integrate from θ = 0 to θ = π. For r, we can integrate from r = 0 to r = 6sinθ.

Putting it all together, the integral expression for the area is A = ∫0π∫06sinθ (6sinθ)·dr·dθ.

Answer 2

r2=x2+y2 and sin(θ)=y/r Therefore, r=SQRT(x2+y2) and 6sin(θ)=6yr The original equation turns into:
SQRT(x2+y2)=6y/ror x2+y2=6y Completing the square turns this into x2+(y−3)^2=9, a circle centered at (0,3) with a radius of 3 The area of that circle is 9π