Question

The surface of an island is defined by the following function over the region on which the function is non negative. Find the volume V of the island shown to the right. z=e−(x2+y2)/4−e−0 Set up the double intregation in polar coordinates, that is used to find the volume ∬ distance

Answer 1

To find the volume of the island, set up a double integral in polar coordinates by expressing the given function in polar form. The conversion from Cartesian to polar coordinates is needed and then substitute the expressions into the given function. Setup the double integral using the values for r and θ to find the volume of the island.

Step-by-step explanation:

To set up the double integral in polar coordinates to find the volume of the island, we need to express the given function in polar form. The conversion from Cartesian coordinates (x, y) to polar coordinates (r, θ) is given by:

x = r * cos(θ)

y = r * sin(θ)

Substituting these expressions into the given function z = e^(-(x^2 + y^2)/4) - e^(-0), we get:

z = e^(-(r^2 * cos^2(θ) + r^2 * sin^2(θ))/4) - 1

Now we can set up the double integral in polar coordinates to find the volume V:

V = ∬ z * r dr dθ

Note: Please provide the limits of integration for r and θ to complete the calculation.