Question

Tutorial Exercise: Evaluate the line integral, where (C) is the given curve.

∫C xy^2 , ds, C is the right half of the circle x^2 + y^2 = r^2, text with 0 ≤ x ≤ r, , y ≥ 0.
a) ( π r^416 )
b) ( π r^48 )
c) ( π r^432 )
d) ( π r^44 )

Answer 1

The question involves evaluating a line integral of xy^2 over the upper half of a circle in polar coordinates, which involves setting up the integral in terms of polar angle θ and then using trigonometric identities to simplify and solve it.

Step-by-step explanation:

The task is to evaluate the line integral of the function xy^2 along the curve C, which is the right half of the circle x^2 + y^2 = r^2 with 0 ≤ x ≤ r and y ≥ 0. To compute this, we parametrize the curve using polar coordinates x = r cos(θ), y = r sin(θ) where the angle θ ranges from 0 to π/2 for the upper half of the circle.

We then need to calculate the differential arc length, ds, which is given by the formula ds = sqrt((dx/dθ)^2 + (dy/dθ)^2) dθ, for the curve in polar coordinates. Substituting our parametrization, we get ds = sqrt((-r sin(θ))^2 + (r cos(θ))^2) dθ = r dθ since x^2 + y^2 = r^2.

The integral then becomes ∫C xy^2 ds = ∫_0^π/2 (r^3 cos(θ) sin^2(θ)) r dθ = r^4 ∫_0^π/2 cos(θ) sin^2(θ) dθ. We can use a power-reduction formula for sin^2(θ) to convert it to a form easier to integrate, resulting in the final evaluation of the integral as a fraction of π multiplied by some power of r.