Find all points of intersection (r,θ) of the curves r=2cos(θ), r=4sin(θ).
Note: In this problem the curves intersect at the pole and one other point. Only enter the answer for nonzero r in the form (r,θ) with θ measured in radians.
POINT OF INTERSECTION: (4/sqrt5, atan(1/2))
Given the point of intersection above what is the area?? I have been stuck working on this for awhile and many tutors can not get the right area...
per the usual formula,
A = ∫ 1/2 r^2 θ dθ
but the boundary changes at Ø = arctan(1/2) so
A1 = ∫[0,arctan(1/2)] 1/2 (4sinθ)^2 dθ = 4Ø - 8/5
A2 = ∫[arctan(1/2),π/2] 1/2 (2cosθ)^2 dθ = π/2 - 2/5 - Ø
A = A1 + A2 = π/2 - 2 + 3Ø = 0.9617
check using rectangular coordinates; we have two circles
(x-1)^2 + y^2 = 1
x^2 + (y-2)^2 = 4
They intersect at (8/5, 4/5) so the area is
∫[0,8/5] √(1 - (x-1)^2) - (2-√(4-x^2)) dx = 0.9617
To find the area between the curves, we need to find the points of intersection first. In this case, we have two polar curves: r = 2cos(θ) and r = 4sin(θ).
Step 1: Find the points of intersection by equating the two equations:
2cos(θ) = 4sin(θ)
Step 2: Simplify the equation:
cos(θ) = 2sin(θ)
Step 3: Divide both sides by sin(θ):
cot(θ) = 2
Step 4: Find the values of θ where cot(θ) = 2:
To solve this, we can take the inverse cotangent of both sides:
θ = cot^(-1)(2)
Step 5: Calculate the value of θ using a calculator:
θ ≈ 0.4636 radians
Step 6: Find the corresponding value of r for each θ using either of the original equations:
r = 2cos(θ)
r ≈ 2cos(0.4636)
r ≈ 1.8478
Step 7: Convert the polar coordinates (r, θ) into rectangular coordinates (x, y):
For the point of intersection (r, θ) ≈ (1.8478, 0.4636), we can convert it to rectangular coordinates as follows:
x = r * cos(θ) ≈ 1.8478 * cos(0.4636) ≈ 1.6513
y = r * sin(θ) ≈ 1.8478 * sin(0.4636) ≈ 0.7960
Therefore, the point of intersection in rectangular coordinates is approximately (1.6513, 0.7960).
Now that we have the point of intersection, let's calculate the area enclosed by the curves.
Step 1: Draw the graph of the curves to visualize the region in the polar coordinate system.
Step 2: Find the limits of integration for the angle θ. Since the curves intersect at the pole (θ = 0) and at the point (r, θ) ≈ (1.8478, 0.4636), the limits of integration for θ are 0 and 0.4636 radians.
Step 3: Set up the integral for the area using the formula for the area between two curves in polar coordinates:
Area = (1/2) ∫[θ1,θ2] (r1^2 - r2^2) dθ
Here, r1 and r2 are the given polar curves, and θ1 and θ2 are the lower and upper limits of integration.
Step 4: Plug in the values into the integral:
Area = (1/2) ∫[0,0.4636] [(2cos(θ))^2 - (4sin(θ))^2] dθ
Step 5: Simplify the expression inside the integral:
Area = (1/2) ∫[0,0.4636] [4cos^2(θ) - 16sin^2(θ)] dθ
Step 6: Integrate the expression with respect to θ:
Area = (1/2) [4θ + 8sin(2θ)] evaluated from 0 to 0.4636
Step 7: Calculate the values of the integral:
Area = (1/2) [(4 * 0.4636) + 8sin(2 * 0.4636) - 0]
Step 8: Evaluate the expression:
Area ≈ 1.2287 square units
Therefore, the approximate area enclosed by the curves is 1.2287 square units.