Answer:
V = 6.148
Step-by-step explanation:
1. We express the elliptic paraboloid function in terms of z:
[tex]z = 1-\frac{x^{2} }{4}-\frac{y^{2} }{9}[/tex]
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:
[tex]V = 4\int\limits^1_0\int\limits^2_0 {1-\frac{x^{2} }{4}-\frac{y^{2} }{9}} \, dydx[/tex]
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:
[tex]V = 4\int\limits^1_0 {\frac{46}{27}-\frac{x^{2} }{2}} \, dx[/tex]
4. We solve the last integral to get:
[tex]V = 6.148[/tex]
