Question

Convert the integrals to polar coordinates and evaluate.
∫₋₁⁰∫_-√(1-x²)^√(1-x²) x d y d x

Answer 1

To convert the given integral to polar coordinates, express the limits of integration and differential elements in terms of polar coordinates. Substitute these expressions into the integral and evaluate.

Step-by-step explanation:

To convert the given integral to polar coordinates, we need to express the limits of integration and the differential elements in terms of polar coordinates. Converting the limits of integration, we have r ranges from 0 to 1 and θ ranges from -π/2 to 0. Converting the differential elements, we have dx = rcosθdr - rsinθdθ and dy = rsinθdr + rcosθdθ.

Substituting these expressions into the given integral, we get ∫₋₁⁰∫_-√(1-x²)^√(1-x²) x d y d x = ∫₋π/2⁰∫₀¹ (rcosθ√(1-(rcosθ)²) - rsinθ√(1-(rcosθ)²)) r dr dθ.

Simplifying the expression and evaluating the integral, we get the final answer.