Find the volume of the solid that lies under the elliptic paraboloid x2/4 + y2/9 + z = 1 and above the rectangle R = [−1, 1] × [−2, 2].

V = 6.148

Step-by-step explanation:

1. We express the elliptic paraboloid function in terms of z:

2. We need to find the volume above the plane R, therefore, we create a double integral for z with the limits described by R:

Note that the limits of the integral start in 0 an not in -2 and -1, also note the number four multiplying the integral, we are using the symmetric property of the elliptic parable to find a portion of the volume, in different sections of the geometry the volume is the same, so we find a portion of the volume and multiply by four to get the total volume (For more information refer to symmetric double integrals)

3. We solve the first integral to get:

4. We solve the last integral to get:

look at the picture.

y-intercept for the function on a picture is 4 → point (0, 4).

y-intercept for the function

is for x = 0. substitute:

y-intercept is 8 → point (0, 8).

.95

step-by-step explanation: