Question

Evaluate the integral by making an appropriate change of variables. ∫∫ r 9 cos⁷(y - x) y x da where r is the trapezoidal region with vertices (1, 0), (3, 0), (0, 3), and (0, 1)?

Answer 1

To evaluate the given integral, use an appropriate change of variables to polar coordinates. The new integral becomes an expression in terms of r and θ. Simplify the expression and apply the limits of integration for r and θ.

Step-by-step explanation:

To evaluate the integral ∫∫ r 9 cos⁷(y - x) y x dA, the appropriate change of variables is to use polar coordinates.A recursive rule requires the previous term to find the current term, whereas an explicit formula provides a direct calculation for any term based on its position in the sequence In polar coordinates, the region r is defined by the inequalities 1 ≤ r ≤ 3 and 0 ≤ θ ≤ π/2. The new integral becomes ∫∫ 9cos⁷(r sin(θ) - r cos(θ)) (r sin(θ)) (r cos(θ)) r dr dθ.

After expanding the expression and simplifying, the integral becomes:

∫∫ 9r¹⁰ cos⁷(θ) sin²(θ) cos(θ) dr dθ.

The limits of integration are 1 to 3 for r and 0 to π/2 for θ.