Area Bounded by Polar Curves
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=12⁢r2⁢θ.
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, ... , θn−1, b = θn and equal width Δθ.
Consider the i th subinterval θi−1, θi and choose some θi∗ ∈ θi−1, 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=12⁢rθi∗2⁢Δθ.
Therefore, an approximation of the entire area is: A ≈ ∑i=1nAi = ∑i=1n12⁢rθi∗2⁢Δθ.
Allowing the width of each subinterval to become infinitely small by letting n approach infinity, we obtain A = limn→∞∑i=1n12⁢rθi∗2⁢Δθ = ∫ab12⁢rθi∗2⁢ ⅆθ.
Thus, A=12⁢∫abr⁡θ2ⅆθ 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 = 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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