2 min readβ’april 16, 2020

Sumi Vora

When we calculate the **area under the curve for Cartesian graphs**, we would integrate with rectangles, since it is a rectangular plane. However, when we find the area under the curve for polar functions, we need to add up the area of triangles (imagine cutting a pizza into really really thin slices).Β

Since the area of a triangle is calculated by (1/2)bh, where h = r and b = rdΞΈ (the base would be proportional to the radius, multiplied by the tiny value d to obtain an infinitely tiny base).

The tricky part about calculating the area is finding the interval on which you want to integrate. Sometimes, they will give you the graph of the function, or you will be able to graph it on your calculator. However, for non-calculator sections, you might have to figure out the endpoints just with the function.Β

- If the equation has a sinΞΈ or cosΞΈ without any squares or coefficients, then the interval will always be from 0 to 2Ο, since it will go around a full circle once within that intervalΒ
- If the equation has a coefficient inside the trig function (sin4ΞΈ or cos2ΞΈ, etc), then it the function likely has multiple petals. You can make a chart in order to find where the function meets zero, and then just calculate the area of one petal and multiply it by the number of petals.Β
- If you canβt figure out the boundaries, you can usually guess that they are 0 to 2Ο. However, you should only use this as a last resort.Β

First, since there is a coefficient inside of the sine function, we can assume that there will be petals to the functionΒ

We can figure out the length of one petal by making a chart:

We can see that this pattern will continue; the graph will come back to the origin 8 times over [0, 2), so there are 8 petalsΒ

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