Workshop 7

Workshop 7 - The function r = 1 sin can be graphed in the r...

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The function r = 1 - sin θ can be graphed in the r, θ plane to help graph the cardioid in the x, y plane. The graph of the function in the r, θ plane can be derived by evaluating 1 - sin θ for various θ ’s from 0 to 2 π . By seeing how 1-sin θ behaves in the r, θ plane, the function of 1-sin θ can be graphed in the x, y plane. To graph the cardioid, start from θ = 0 and plot increments of π /2 all the way to 2 π , which is derived from the graph above ( θ = 0, r = 1; θ = π /2, r = 0; θ = π , r = 0; θ = 3 π /2, r = 2; θ = 2 π , r = 1). Essentially, by making θ range from 0 to 2 π , it is possible to graph one period of r = 1 - sin θ , which can be derived from the graph in the x, y plane. In addition to graphing the points in increments of π /2, it must be observed that the r (of r = 1 - sin θ ) decreases from θ = 0 to θ = π /2, increases from θ = π /2 to θ = π , increases from θ = π to θ = 3 π /2, and then decreases from θ = 3 π /2 to θ = 2 π , to trace out the graph of the cardioid. Besides graphing the cardioid, the function r = 1 should be graphed; this function is just a circle with radius one, with the center at the origin, in the x, y plane. The first part of the problem requires finding what percent of the cardioids enclosed area lies above the x-axis. To do this the area of the cardioid from 0 to π (which lies above the x axis) must be divided by the entire area of the cardioid from 0 to 2
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This note was uploaded on 02/29/2012 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Workshop 7 - The function r = 1 sin can be graphed in the r...

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