Find the area of the region which is bounded by the polar curves
theta =pi and
r=2theta 0<theta<1.5pi inclusive
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To find the area bounded by polar curves, we can use the formula for the area of a region in polar coordinates, which is given by:
A = (1/2)∫[a, b] (r(θ))^2 dθ
In this case, we need to find the area bounded by the curves θ = π and r = 2θ for 0 ≤ θ ≤ 1.5π.
Let's start by finding the limits of integration.
The given condition is 0 ≤ θ ≤ 1.5π inclusive, which means we need to integrate the area from θ = 0 to θ = 1.5π.
Now, let's find the equation for r in terms of θ. The polar curve is given by r = 2θ.
To express r in terms of θ, we substitute the given equation into (r(θ))^2:
(r(θ))^2 = (2θ)^2 = 4θ^2
Now, we can plug the values into the formula for the area:
A = (1/2)∫[0, 1.5π] (4θ^2) dθ
To evaluate this integral, we can simplify it further by multiplying the coefficient inside the integral:
A = 2∫[0, 1.5π] (θ^2) dθ
Now, we need to evaluate the integral of θ^2 with respect to θ.
∫(θ^2) dθ = (1/3)θ^3 + C
Where C is the constant of integration.
Now, we can substitute the limits of integration and calculate the area:
A = 2[(1/3)(1.5π)^3 - (1/3)(0)^3]
Simplifying further:
A = 2(3π^3/3)
A = 2π^3
Therefore, the area bounded by the polar curves θ = π and r = 2θ for 0 ≤ θ ≤ 1.5π inclusive is 2π^3 square units.