Question

Using polar coordinates, evaluate the improper integral ∫∫²^(−7(x² + y²)) x y.

Answer 1

To evaluate the improper integral using polar coordinates, convert the given function from rectangular coordinates to polar coordinates. Find the limits of integration for r and θ, and then evaluate the integral.

Step-by-step explanation:

To evaluate the improper integral using polar coordinates, we first need to convert the given function from rectangular coordinates to polar coordinates. In polar coordinates, x = rcosθ and y = rsinθ.

So, the function becomes f(r, θ) = r^(-7(r^2cos^2θ + r^2sin^2θ))rcosθrsinθ.

Next, we need to find the limits of integration for r and θ. For r, the limits are from 0 to ∞, and for θ, the limits are from 0 to 2π.

Now we can evaluate the integral ∫∫r^(-7(r^2cos^2θ + r^2sin^2θ))rcosθrsinθ, with r ranging from 0 to ∞, and θ ranging from 0 to 2π.