Question

Consider the solid bounded by z = 4 − x, x = 6 − 4y, x = 4y, x = 0, and z = 0 with density function δ(x,y,z) = 38xy kg/m3 and length measured in meters. Evaluate the inner integrals by hand and then use your calculator/technology to evaluate the third integral. Round your answers to 2 decimal places.

Answer 1

To evaluate the inner integrals in the given problem, we first find the limits of integration for x, y, and z. Then, we set up the triple integral and integrate with respect to z, y, and x in that order. Finally, we evaluate the triple integral using a calculator or technology to obtain the numerical result.

Step-by-step explanation:

First, let's find the limits of integration for the inner integrals.

From the equation x = 0, we know that the lower limit for x is 0.

From the equation x = 6 - 4y, we can rewrite it as y = (6 - x)/4 and find that x can go from 0 to 6.

From the equation x = 4y, we can rewrite it as y = x/4 and find that x can go from 0 to 4.

From the equation z = 4 - x, we can rewrite it as x = 4 - z and find that z can go from 0 to 4.

So, the limits of integration for the inner integrals are:

1. x: 0 to 4
2. y: (6 - x)/4 to x/4
3. z: 0 to 4 - x

Next, let's set up the triple integral:

∫∫∫ 38xy dz dy dx

To evaluate this triple integral, we integrate with respect to z first, then with respect to y, and finally with respect to x.

Integrating with respect to z:

∫ (38xyz) | from 0 to 4 - x = 38xy(4 - x)

Integrating with respect to y:

∫ (38xy(4 - x)) | from (6 - x)/4 to x/4 = 38[(6 - x)/4](x/4)(4 - x)

Integrating with respect to x:

∫ 38[(6 - x)/4](x/4)(4 - x) | from 0 to 4