Question

et r be a region that is enclosed by the circle x2 y2 = 2x. sketch the region and writerr r f(x,y)da as an iterated integral in polar coordinates.

Answer 1

The circle x^2 + y^2 = 2x shifts to polar coordinates as r = 2cos(θ) for r ≥ 0. The integral ∫∫_r f(x,y)da transforms as ∫0^2π ∫0^2cos(θ) f(rcos(θ), rsin(θ))r dr dθ in polar coordinates.

Step-by-step explanation:

The equation is given by x^2 + y^2 = 2x. To convert this equation to polar coordinates, we know that x = rcos(θ) and y = rsin(θ). Substituting the polar coordinates into the equation, we get r^2 = 2rcos(θ), or r = 2cos(θ) for r ≥ 0. This is the equation of a circle in polar coordinates, with center at (1,0) and radius 1. The region enclosed by this circle in the xy-plane is the region r, and the iterated integral ∫∫_r f(x,y)da in polar coordinates can be written as ∫0^2π ∫0^2cos(θ) f(rcos(θ), rsin(θ))r dr dθ. The factor ‘r’ is necessary because the differential area element in polar coordinates is rdrdθ instead of dxdy.

Learn more about Coordinate Change