Question

By changing to polar coordinates, evaluate the integral

∬D(x2+y2)¹¹/²dA, where D is the disk x²+y²≤9
By changing to polar coordinates, evaluate the integral ∬D(x²+y²)¹¹/² d A where D is the disk x²+y²≤ 9.
The value is___

Answer 1

To evaluate the given integral in polar coordinates, substitute x and y in terms of r and θ. Express the integral in terms of polar variables r and θ. Integrate the resulting expression over the specified limits of r and θ to obtain the value of the integral.

Step-by-step explanation:

To evaluate the given integral in polar coordinates, we need to express the integral in terms of polar variables.

We know that x = r*cos(θ) and y = r*sin(θ), where r is the radial distance and θ is the angular distance.

Substituting these expressions into the integrand, we have (x²+y²)¹¹/² = (r²*cos²(θ) + r²*sin²(θ))¹¹/² = r¹¹/².

The integral then becomes ∬D(x²+y²)¹¹/² dA = ∬D r¹¹/² * 2πr dr dθ.

Applying the formula for the area element dA in polar coordinates, we can rewrite the integral as ∫∫D r¹¹/² * 2πr dr dθ.

Next, we integrate with respect to r from 0 to 3 (the limits of integration for r) and with respect to θ from 0 to 2π (the complete revolution of θ).

Performing the integration, we obtain the value of the integral as $(32π)/15$.