Final answer:
To evaluate the given improper integral using polar coordinates, convert the integral to polar form, perform a substitution to simplify, and then integrate over the radial and angular components to arrive at the final solution, which is π.
Step-by-step explanation:
The student is asking to evaluate the improper integral ∫∫R^2 e^{-7(x^2 +y^2)} dxdy using polar coordinates. To do this, we first express the integral in terms of polar coordinates, where x = rcos(θ) and y = rsin(θ). We also need to change the differential area element dxdy to polar coordinates, which becomes r dr dθ. The integral will be from r=0 to r=∞ (as it's an improper integral over the entire plane) and θ=0 to θ=2π. The exponential function simplifies to e^{-7r^2} when expressed in polar coordinates:
∫∫R^2 e^{-7(x^2 +y^2)} dxdy = ∫_{0}^{2π} dθ ∫_{0}^{∞} r e^{-7r^2} dr
Now, we can evaluate the radial integral first. We substitute u = 7r^2 and du = 14r dr. After adjusting the limits and the differential, we integrate u from 0 to ∞:
∫_{0}^{∞} r e^{-7r^2} dr = ½ ∫_{0}^{∞} e^{-u} du = -½ e^{-u} |_{0}^{∞} = ½
The angular integral is straightforward as it is just an integral over θ from 0 to 2π, which equals 2π. Hence, the final answer is 2π * ½ = π.