Page 56 - Calculus for the Clueless
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Symmetry is about x-axis, y-axis, and the origin. So we only need to graph the first quadrant and then repeat the
curve perfectly in the other three. However, since we have 2θ, we need 3 columns—2θ, θ, and r—with only the
last two on the graph.
2 θ θ r
0 0 3
π/6 π/12 2.8
π/4 π/8 2.5
π/3 π/6 2.1
π/2 π/4 0
Although this problem can be done without a calculator if you can approximate well, it is convenient to use a
calculator for this one.
We would now like to find the area in polar coordinates. In almost any calc book, you will find the area is
Example 8—
Find the area of the cardioid r = 2 + 2 cos θ.
As we have seen before, this curve is symmetric with respect to the x-axis. We can integrate from 0 to 2π or
double the integral from 0 to π.
Example 9—
