Evaluate the triple integral ∫∫∫_E (x^2.e^y)dV where E is bounded by the parabolic cylinder z=1−y^2 and the planes z=0, x=1 and x=−1.

To evaluate the triple integral ∫∫∫_E (x^2.e^y)dV, where E is bounded by the parabolic cylinder z=1−y^2 and the planes z=0, x=1, and x=−1, we need to set up the integral using appropriate limits of integration.

First, we need to determine the boundaries of integration for each variable: x, y, and z.

Since we are given that E is bounded by the planes x=1 and x=−1, the limits of integration for x will be -1 to 1.

Next, let's determine the limits of integration for y. The parabolic cylinder z=1−y^2 intersects the xy-plane at y=1 and y=-1. Therefore, the limits of integration for y will be -1 to 1.

Finally, we need to determine the limits of integration for z. The parabolic cylinder z=1−y^2 is bounded by the plane z=0, so the limits of integration for z will be 0 to 1−y^2.

The triple integral is now set up as follows:

∫∫∫_E (x^2.e^y) dV

∫[-1 to 1] ∫[-1 to 1] ∫[0 to 1-y^2] (x^2.e^y) dz dy dx

Now we can evaluate the integral step by step.

First, let's integrate with respect to z:
∫[0 to 1-y^2] (x^2.e^y) dz = (x^2.e^y) * z |[0 to 1-y^2]
= (x^2.e^y) * (1-y^2) - (x^2.e^y) * 0
= (x^2.e^y) * (1-y^2)

Next, let's integrate with respect to y:
∫[-1 to 1] (x^2.e^y) * (1-y^2) dy
= (x^2.e^y) * (1-y^2) * y |[-1 to 1]
= (x^2.e^1) * (1-1^2) * 1 - (x^2.e^-1) * (1-(-1)^2) * (-1)
= 2x^2.e^1

Finally, let's integrate with respect to x:
∫[-1 to 1] 2x^2.e^1 dx
= 2.e^1 * (x^3/3) |[-1 to 1]
= 2.e^1 * (1^3/3) - 2.e^1 * ((-1)^3/3)
= 2.e^1 * (1/3 + 1/3)
= 4/3.e^1

Therefore, the value of the triple integral ∫∫∫_E (x^2.e^y)dV is 4/3.e^1.

To evaluate the given triple integral ∫∫∫E (x^2 * e^y) dV, where E is bounded by the parabolic cylinder z=1−y^2 and the planes z=0, x=1, and x=−1, we can use the method of cylindrical coordinates.

Cylindrical coordinates are defined by (r, θ, z), where r is the radial distance from the origin, θ is the azimuthal angle in the xy-plane, and z is the vertical coordinate.

First, let's determine the limits of integration for each variable:

1. The limits for r: Since there are no restrictions on the radial distance, r can range from 0 to infinity.

2. The limits for θ: Since the problem does not provide any limitations regarding the azimuthal angle, θ can vary from 0 to 2π (a complete revolution around the z-axis).

3. The limits for z: The parabolic cylinder is bounded by the planes z=0, z=1−y^2, x=1, and x=−1. Since the cylinder extends infinitely along the z-axis, the limits for z extend from 0 to 1−y^2.

Now, let's set up the triple integral using cylindrical coordinates:

∫∫∫E (x^2 * e^y) dV = ∫(0 to 2π) ∫(0 to ∞) ∫(0 to 1−y^2) (r^2 * e^y) dz dr dθ

Next, evaluate the innermost integral with respect to z:

∫(0 to 1−y^2) (r^2 * e^y) dz = (r^2 * e^y) * (1−y^2)

Now, integrate with respect to r:

∫(0 to ∞) (r^2 * e^y) * (1−y^2) dr

Finally, integrate with respect to θ:

∫(0 to 2π) ∫(0 to ∞) (r^2 * e^y) * (1−y^2) dr dθ

At this point, you can evaluate the remaining integrals numerically (if possible) or leave the result in the form of a triple integral.