Evaluate the following integral using three different orders of integration.
*triple integral E (xz − y3) dV,
where E =
(x, y, z) | −1 ≤ x ≤ 3, 0 ≤ y ≤ 4, 0 ≤ z ≤ 3
To evaluate the given triple integral using different orders of integration, we will need to switch the order of integration for each evaluation. Let's consider three different orders of integration: x, y, z; y, x, z; and z, y, x.
1. x, y, z order:
We start by integrating with respect to x first, then y, and finally z.
∫∫∫E (xz − y^3) dV
= ∫(0 to 3) ∫(0 to 4) ∫(−1 to 3) (xz − y^3) dx dy dz
Integrating (xz − y^3) with respect to x gives:
= ∫(0 to 3) [(x^2z/2) − y^3x] |(−1 to 3) dy dz
= ∫(0 to 3) [(9z/2) − 4y^3] dy dz
Integrating (9z/2) − 4y^3 with respect to y gives:
= ∫(0 to 3) [(9z/2)y − y^4] |(0 to 4) dz
= ∫(0 to 3) [(18z − 64) / 2] dz
= (18z^2 - 64z) / 4 |(0 to 3)
= (162 - 192)/4
= -7/2
Therefore, the value of the triple integral using the order x, y, z is -7/2.
2. y, x, z order:
Next, we will integrate with respect to y first, then x, and finally z.
∫∫∫E (xz − y^3) dV
= ∫(−1 to 3) ∫(0 to 3) ∫(0 to 4) (xz − y^3) dy dx dz
Integrating (xz − y^3) with respect to y gives:
= ∫(−1 to 3) [(xy - y^4/4) ] |(0 to 4) dx dz
= ∫(−1 to 3) [(4x - 256/4) ] dx dz
= ∫(−1 to 3) (4x - 64) dx dz
= 2x^2 - 64x |(−1 to 3)
= (18 - 192) - (-2 + 64)
= -160
Therefore, the value of the triple integral using the order y, x, z is -160.
3. z, y, x order:
Lastly, we will integrate with respect to z first, then y, and finally x.
∫∫∫E (xz − y^3) dV
= ∫(−1 to 3) ∫(0 to 4) ∫(0 to 3) (xz − y^3) dz dy dx
Integrating (xz − y^3) with respect to z gives:
= ∫(−1 to 3) [(xz^2/2) − y^3z] |(0 to 3) dy dx
= ∫(−1 to 3) [(9x/2) − 27y^3] dy dx
Integrating (9x/2) − 27y^3 with respect to y gives:
= ∫(−1 to 3) [(9x/2)y - 9y^4] |(0 to 4) dx
= ∫(−1 to 3) [(18x - 1152)/2] dx
= 9x^2 - 1152x |(−1 to 3)
= 243 - 3456 - (-9 + 1152)
= -1968
Therefore, the value of the triple integral using the order z, y, x is -1968.
In summary,
Triple integral using x, y, z order: -7/2
Triple integral using y, x, z order: -160
Triple integral using z, y, x order: -1968