find the area inside a single petal of r=5cos2theta
I did one just about like this. What do you think you need to do?
2integral[π/4-0](5cos2theta)^2dtheta
=25π/8
looks good to me
except it's A = integral 1/2 r^2 dtheta
To find the area inside a single petal of the polar curve r = 5cos(2θ), you can follow these steps:
Step 1: Graph the polar curve:
Start by sketching a graph of the polar curve in the polar coordinate plane. The curve r = 5cos(2θ) represents a flower with eight petals symmetric about the origin.
Step 2: Identify the bounds for θ:
In this case, each petal spans a certain range of angle values. To find the area of one petal, you need to determine the limits of integration for θ. Looking at the graph, one petal corresponds to an angle range of π/4 to 3π/4 (or 45° to 135°).
Step 3: Set up the integral:
To calculate the area, you will integrate over the given range of θ. The general formula for finding the area inside a polar curve is:
A = (1/2) ∫[θ1, θ2] r² dθ,
where r is the given polar equation and θ1 and θ2 are the bounds for θ.
In this case, the area formula becomes:
A = (1/2) ∫[π/4, 3π/4] (5cos(2θ))² dθ.
Step 4: Evaluate the integral:
Now, integrate the expression inside the integral to evaluate the area. Simplify the integrand, expand using the double-angle formula, and then integrate:
A = (1/2) ∫[π/4, 3π/4] (25cos²(2θ)) dθ
= (1/2) ∫[π/4, 3π/4] (25(1 + cos(4θ))/2) dθ
= (25/4) ∫[π/4, 3π/4] (1 + cos(4θ)) dθ.
Evaluating this integral will give you the area inside one petal of the polar curve r = 5cos(2θ).