Question

Evaluate ∭_ eˣ+y+z d V for the specified region .
: 0 ≤ x ≤ 5, 0 ≤ y ≤ x, 0 ≤ z ≤ 13

Answer 1

To evaluate the triple integral ∭ eˣ+y+z dV for the specified region 0 ≤ x ≤ 5, 0 ≤ y ≤ x, 0 ≤ z ≤ 13, we need to integrate the function eˣ+y+z with respect to x, y, and z over the given limits of integration.

Step-by-step explanation:

To evaluate the triple integral ∭ eˣ+y+z dV for the region defined by 0 ≤ x ≤ 5, 0 ≤ y ≤ x, and 0 ≤ z ≤ 13, we need to integrate the function eˣ+y+z with respect to x, y, and z over the given limits of integration.

1. First, we integrate with respect to x: ∫0ˆ5 eˣ+y+z dx = eˣ+y+z ∫0ˆ5 dx = eˣ+y+z [x]ˆ50 = eˣ+y+z (5 - 0) = 5eˣ+y+z.
2. Next, we integrate the result from step 1 with respect to y: ∫0ˆx 5eˣ+y+z dy = 5eˣ+y+z ∫0ˆx dy = 5eˣ+y+z [y]ˆx0 = 5eˣ+y+z (x - 0) = 5xeˣ+y+z.
3. Finally, we integrate the result from step 2 with respect to z: ∫0ˆ13 5xeˣ+y+z dz = 5xeˣ+y+z ∫0ˆ13 dz = 5xeˣ+y+z [z]ˆ130 = 5xeˣ+y+z (13 - 0) = 65xeˣ+y+z.

Therefore, the evaluated triple integral is 65xeˣ+y+z for the given region 0 ≤ x ≤ 5, 0 ≤ y ≤ x, and 0 ≤ z ≤ 13.