Find the area of the region which is bounded by the polar curves
theta =pi and
r=2theta 0<theta<1.5pi inclusive
To find the area of the region bounded by the polar curves, we can use the formula for calculating the area enclosed by a polar curve.
The formula is given by:
A = (1/2) ∫[a, b] (r^2) dθ
where A represents the area of the region, a and b are the angles of rotation, and r represents the polar curve.
In this case, we are given the polar curves as follows:
Curve 1: θ = π and
Curve 2: r = 2θ, where 0 < θ < 1.5π (inclusive)
Firstly, we need to find the limits of integration, which are the angles of rotation. From the given conditions, curve 1 starts at θ = π and curve 2 ends at θ = 1.5π. Therefore, our limits of integration become a = π and b = 1.5π.
Now we substitute the given equation of curve 2, which is r = 2θ, into the formula for calculating the area:
A = (1/2) ∫[π, 1.5π] (2θ)^2 dθ
Simplifying the equation, we get:
A = (1/2) ∫[π, 1.5π] 4θ^2 dθ
To solve this definite integral, we need to use the power rule of integration.
The power rule states that the integral of x^n with respect to x is equal to x^(n+1)/(n+1), where n is not equal to -1.
Now applying the power rule to our integral:
A = (1/2) [4θ^3/3] [π, 1.5π]
Substituting the limits of integration, we have:
A = (1/2) [(4(1.5π)^3/3) - (4π^3/3)]
Calculating further:
A = (1/2) [54π^3/3 - 4π^3/3]
Simplifying:
A = (1/2) [50π^3/3]
Finally, we can simplify the expression by dividing both the numerator and denominator by 2 to obtain:
A = 25π^3/3
Therefore, the area of the region bounded by the given polar curves is 25π^3/3 square units.