Question

Find the double integral in polar coordinates that describes the volume of the region that lies under the paraboloi=ʸ² and above the triangle enclosed by the lines y=x,x=0 and x+y=2 in the xy-plane. No need to solve the integral.d z=ˣ²

Answer 1

To find the double integral in polar coordinates that describes the volume of the region, we need to determine the limits of integration in polar form and set up the integral with respect to r and theta. The integrand will be the function dz = r^2.

Step-by-step explanation:

To find the double integral in polar coordinates that describes the volume of the region, we first need to determine the limits of integration in polar form. The triangle enclosed by the lines y=x, x=0, and x+y=2 can be rewritten in polar form as r cos(theta) = r sin(theta) and r cos(theta) + r sin(theta) = 2. Simplifying these equations, we get r = 2 / (cos(theta) + sin(theta)) and r = 2 cos(theta), respectively. The limits of integration for r will be the distances between these two curves.

Once we have the limits of integration in polar form, we can set up the double integral. The integral for the volume will have two components: the integral with respect to r and the integral with respect to theta. The integrand will be the function dz = r^2, and the limits of integration will be the limits we calculated earlier.

So the double integral in polar coordinates that describes the volume of the region is ∫∫ (r^2) dr d(theta), where the limits of integration are determined by r = 2 / (cos(theta) + sin(theta)) and r = 2 cos(theta).