The value of the triple integral ∭ eˣ⁺ʸ⁺ᶻ²dV over the region E enclosed by the ellipsoid z² + x²/4 + y²/9 = 1 and the surface z = x²/4 + y²/9 is approximately 1.79 cubic units.
In this context, the triple integral ∭ eˣ⁺ʸ⁺ᶻ²dV represents the volume calculation over the region E, which is the solid enclosed by the given ellipsoid and the surface z = x²/4 + y²/9.
The ellipsoid is defined by the equation z² + x²/4 + y²/9 = 1, indicating a three-dimensional shape. Additionally, the surface z = x²/4 + y²/9 contributes to the boundary of the region E.
To calculate the volume within this region, the triple integral integrates the function eˣ⁺ʸ⁺ᶻ² over the specified ellipsoidal space. The integral considers variations in x, y, and z within the given constraints.
The task involves visualizing the geometry of E, where the ellipsoid and the surface intersect to enclose a three-dimensional region. Sketching the graph of E is helpful, and labeling each surface provides a clear representation of the solid.
Furthermore, sketching the projection onto the xy-plane of the curve of intersection between the surfaces z = √(x²/4 + y²/9) and z = x²/4 + y²/9 helps to visualize the overlap in the x-y plane.