Question

Find the volume of the solid enclosed by the paraboloid z=2+x²+(y−2)²

and the planes z=1,x=1, x=−1,y=0, and y=4

Answer 1

The volume of the solid enclosed by the given paraboloid and planes is found by setting up a double integral over the bounded rectangular region in the xy-plane, and solving it.

Step-by-step explanation:

To find the volume of the solid enclosed by the paraboloid z=2+x²+(y−2)² and the planes z=1, x=1, x=−1, y=0, and y=4, we need to set up an integral over a bounded region in the xy-plane. The bounds for the x values are given to be between −1 and 1, and for the y values between 0 and 4.

Firstly, we note that the volume V we are looking for is above the plane z=1 and below the paraboloid. Therefore, we can use the following integral to find the volume:

V = ∫∫_{D} (2+x²+(y−2)² - 1) dx dy

Where D is the rectangular region described by −1 ≤ x ≤ 1 and 0 ≤ y ≤ 4. This integral can be solved by evaluating the double integral over the bounded region.