CALC BC Lemniscate area bounded by curve

Question

The graph of the lemniscate 𝑟=𝑘𝑐𝑜𝑠2𝜃 where 𝑘 is a constant is shown in the figure to the right. Find the area bounded by the curve.

(A) 𝑘^2/4

(B) 𝑘^2/2

(C) 𝑘^2

(D) 2

Answer 1

When you're getting the area of a polar curve like this one, you will first want to graph the curve. Curves like this have several loops: for even coefficients of θ, this will be 2*A loops and the graph will go between 0 and 2π; for odd coefficients of θ, there will be A loops and the graph will go between 0 and π. In this case, because the coefficient A=2, we will have 4 loops. All these loops are symmetric, so we really only need to do the area of half of one of the loops and then multiply this by 8 for the full area. So we could start at θ=0 and go to θ=π/4 for that "half a loop" area.

Next, for polar coordinates, the area is found by integrating r²/2 on the interval. Noting this, we would get the integral of 4k²cos²(2θ)dθ.

A = 4k² ∫₀^π/4 cos²(2θ)dθ

= 2k² ∫₀^π/4 1 + cos(4θ) dθ --I'm using the double angle formula here to get this in a form we can integrate--

= 2k²(θ + 0.25sin(4θ)) |₀^π/4

= 2k²[(π/4 - 0) + 0.25(sin(π) - sin(0))]

= 2k²(π/4 + 0) --sine=0 for π and 0, so that second term doesn't really mean anything--

= k²π/2

This is the same answer as Kevin achieved, but I'm also noting that this is not in your options.

The same process works for the lemniscate if the formula incorrectly entered and should be r² = k²cos(2θ). Now the function has 2 loops; we can find half of one of these loops by integrating from 0 to π/4. Then multiplying by 4 gives us an area of 2 times the integral of r².

A = 2k² ∫₀^π/4 cos(2θ)dθ

= k²sin(2θ) |₀^π/4

= k²(1-0) = k²

That gives you choice C as the answer.

Answer 2

A = ∫₀^2π0.5k²cos²2θdθ = 0.25k²(θ + sin4θ/4)₀^2π = 0.5πk²

Answer 3

You basically want to compute a double integral, where r ranges from 0 to k*cos(2θ) and where θ ranges from 0 to 2π. You would set this up as:

Area = ∫ [0, 2π] ( ∫ [0, k*cos(2θ)] ( r ) dr ) dθ

which gives an area of πk^2/2.

I'm assuming your answer choices are in fact multiples of π which you are leaving out, so the correct answer in that case would be (B).