How can you find the area of the region determined by 0 ≤ r ≤ f
I have in mind the situation where every ray through the origin having angle θ for θ ∈
This is a representation of a small triangle obtained from two rays whose angles differ by only dθ. What is the area of this triangle, dA? It would be
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with the approximation getting better as the angle gets smaller. Thus the area should solve the initial value problem,
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Therefore, the total area would be given by the integral
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Example 12.2.1 Find the area of the cardioid, r = 1 + cosθ for θ ∈
From the graph of the cardioid presented earlier, you can see the region of interest satisfies the conditions above that every ray intersects the graph in only one point. Therefore, from 12.2 this area is
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Example 12.2.2 Verify the area of a circle of radius a is πa2.
The polar equation is just r = a for θ ∈
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Example 12.2.3 Find the area of the region inside the cardioid, r = 1+cosθ and outside the circle r = 1 for θ ∈
As is usual in such cases, it is a good idea to graph the curves involved to get an idea what is wanted.
The area of this region would be the area of the part of the cardioid corresponding to θ ∈
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This example illustrates the following procedure for finding the area between the graphs of two curves given in polar coordinates.
Procedure 12.2.4 Suppose that for all θ ∈
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