Find all points of intersection (r,theta) of the curves r=4cos(theta), r=1sin(theta). Next find the area inclosed in the intersection of the two graphs.
4cosθ = sinθ
16cos^2θ = 1-cos^2θ
17cos^2θ = 1
cosθ = 1/√17
Now use that value of θ (call it Ø) to get the area
A = ∫[0,Ø] 1/2 (sinθ)^2 dθ
+ ∫[Ø,π/2] 1/2 (4cosθ)^2 dθ
To find the points of intersection between the two curves, we need to set their equations equal to each other and solve for θ.
First, let's set r=4cos(θ) equal to r=1sin(θ):
4cos(θ) = 1sin(θ)
Divide both sides by 4:
cos(θ)/4 = sin(θ)/1
Now we can use the identity tan(θ) = sin(θ)/cos(θ):
cot(θ) = 4
To find the values of θ that satisfy this equation, we can take the inverse cotangent (or arccot) of both sides:
θ = arccot(4)
Using a calculator, we find that arccot(4) ≈ 0.24498 rad (or about 14.04 degrees).
Now, knowing θ, let's plug it back into one of the original equations to find the corresponding values of r. Let's use r=4cos(θ):
r = 4cos(0.24498)
Using a calculator, we find r ≈ 3.972.
So one intersection point is approximately (r=3.972, θ=0.24498).
Now we need to check if there are any other intersections. The curves are periodic, so we can add or subtract multiples of the period to θ to find other solutions. The period of sin(θ) and cos(θ) is 2π. Since we found one intersection point at θ = 0.24498 rad, we can add 2π to θ to find another possible intersection:
θ = 0.24498 + 2π
Using a calculator, we get θ ≈ 6.388 + 2π (approximating π as 3.14159), which is approximately 9.53 rad (or about 546.34 degrees).
Plugging this new value of θ into r=4cos(θ), we find:
r = 4cos(9.53)
Using a calculator, we get r ≈ -0.201.
Therefore, another intersection point is approximately (r=-0.201, θ=9.53).
To find the enclosed area between the two curves, we can calculate the integral of the difference between the two curves as follows:
A = ∫(r₁ - r₂) dθ
where r₁ = 4cos(θ) and r₂ = 1sin(θ).
We need to integrate from the smallest value of θ to the largest value of θ, which we found to be approximately 0.24498 and 9.53, respectively.
So the formula for the area is:
A = ∫(4cos(θ) - 1sin(θ)) dθ, with limits of integration: 0.24498 to 9.53.
Evaluating this integral will give you the enclosed area between the two curves. You can use numerical methods or software to find the exact value of the integral.
Note: Keep in mind that the results are approximate due to rounding.