Question

Evaluate The Triple Integral. 2x DV, Where E = (X, Y, Z) | 0 ≤ Y ≤ 2, 0 ≤ X ≤ 4 − Y2 , 0

Answer 1

To evaluate the triple integral 2x dV, where E = (x, y, z) | 0 ≤ y ≤ 2, 0 ≤ x ≤ 4 − y² , 0, we use the concept of iterated integrals. The triple integral evaluates to 0.

Step-by-step explanation:

To evaluate the triple integral ∫∫∫2x dV, where E = (x, y, z) | 0 ≤ y ≤ 2, 0 ≤ x ≤ 4 − y² , 0, we can use the concept of iterated integrals. The domain E is defined as a region bounded by the xy-plane, a parabolic curve, and the z-axis. We'll start by integrating with respect to x, then y, and finally z.

First, we integrate 2x with respect to x, treating y and z as constants:
∫2x dx = x²

Next, we integrate the resulting expression with respect to y:
∫(x²) dy = x²y

Finally, we integrate the previous result with respect to z, where z ranges from 0 to 0:
∫(x²y) dz = 0

Combining all the integrals, the triple integral evaluates to:
∫∫∫2x dV = 0

Answer 2

To evaluate the triple integral 2x dV, set up the triple integral with the given limits and integrate with respect to X, Y, and Z.

Step-by-step explanation:

To evaluate the triple integral, let's first determine the limits of integration for each variable. We are given that 0 ≤ Y ≤ 2 and 0 ≤ X ≤ 4 - Y². The upper limit for Z is not provided, so we assume it to be 0. Now, let's set up the triple integral:

∫∫∫ 2xdV

The volume element dV can be expressed as dX dY dZ. Therefore, the triple integral becomes:

∫∫∫ 2x dX dY dZ

Next, we integrate with respect to X, Y, and Z using the given limits to evaluate the triple integral.