Area Bounded by Polar Curves - Maple Help

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Area Bounded by Polar Curves

Main Concept

For polar curves of the form r=rθ, the area bounded by the curve and the rays θ = a and θ = b 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 θ is given by A=12r2θ.

We can approximate the area bounded by the polar curve r = rθ and the rays θ = a and θ = b by using sectors of circles.

First, divide a, b up into n subintervals with endpoints a = θ0, θ1, θ2, ... , θn1, b = θn and equal width Δθ.

Consider the i th subinterval θi1, θi and choose some θi  θi1, θi.

The area on this interval is approximately the area of a sector of a circle with central angle Δθ and radius rθi. So,  Ai=12rθi2Δθ.

Therefore, an approximation of the entire area is: A  i=1nAi = i=1n12rθi2Δθ.

Allowing the width of each subinterval to become infinitely small by letting n approach infinity, we obtain A = limni=1n12rθi2Δθ = ab12rθi2 ⅆθ.

Thus, A=12abrθ2ⅆθ is the area of the region bounded by rθ on a  θ  b.

 

 

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.

 

 

rθ = 

Interval θ =  

 

 

The area contained by the curve on this interval is

rθ = 

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