Let x and y denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. that is, their joint density is f(x, y) = 1 π for x 2 + y 2 ≤ 1. find the joint density function of the polar coordinates r = (x2 + y 2 ) 1/2 and θ = atan( y x ).

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step-by-step explanation:

answer:

given the linear inequality :

add 2y to both sides we get;

simplify:

add 12 to both sides of an equation;

simplify:

divide both sides by 2 we get;

or

the related equation for the inequality equation [1] is;

y-intercepts defined as the graph crosses the y-axis.

substitute x =0 in [2] to solve for y;

y = 6

x-intercepts defined as the graph crosses the x-axis.

substitute value y = 0 to solve for x;

subtract 6 from both sides we get

multiply both sides by 2 we get;

or

x = -12

now, plot these point (0, 6) and (-12, 0) on graph as shown below and shade the graph below the solid line.