Area Bounded by Polar Curves - Maple Programming Help

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Area Bounded by Polar Curves

Main Concept

For polar curves of the form $r=r\left(\mathrm{θ}\right)$, the area bounded by the curve and the rays  and  can be calculated using an integral.

 Calculating the Area Bounded by the Curve The area of a sector of a circle with radius r and central angle $\mathrm{θ}$ is given by $A=\frac{1}{2}{r}^{2}\mathrm{\theta }$. We can approximate the area bounded by the polar curve  and the rays  and  by using sectors of circles. First, divide  up into n subintervals with endpoints  and equal width $\mathrm{Δθ}$. Consider the  subinterval  and choose some . The area on this interval is approximately the area of a sector of a circle with central angle $\mathrm{\Delta \theta }$ and radius $r\left({\mathrm{θ}}_{i}^{\ast }\right)$. So,  ${A}_{i}=\frac{1}{2}{\left(r\left({\mathrm{θ}}_{i}^{\ast }\right)\right)}^{2}\mathrm{\Delta \theta }$. Therefore, an approximation of the entire area is: . Allowing the width of each subinterval to become infinitely small by letting n approach infinity, we obtain . Thus, $A=\frac{1}{2}{\int }_{a}^{b}{r\left(\mathrm{\theta }\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆ\mathrm{θ}$ is the area of the region bounded by $r\left(\mathrm{θ}\right)$ on .

Choose a polar function from the list below to plot its graph. Enter the endpoints of an interval, then use the slider or button to calculate and visualize the area bounded by the curve on the given interval. When choosing the endpoints, remember to enter π as "Pi". Note that any area which overlaps is counted more than once.

 1cos(theta)1 - sin(theta)cos(2*theta)sin(3*theta)cos(theta) + sin(theta)3*cos(4*theta)2 - 4*cos(theta)5 + sin(7*theta)exp(cos(4*theta))theta^sin(theta)sqrt(sin(theta)^2) Interval θ =       The area contained by the curve on this interval is



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