Question 1

Which of the following correctly represents the triple integral for finding the volume of a solid region $$E in $$three-dimensional space?

Accepted Answer

$$\int\int\int_E dx \, dy \, dz$$

Question 2

Given the solid in the first octant bounded by the plane $$2x + 3y + z = 12$$ and the coordinate planes, which of the following is the correct set-up for the triple integral to find its volume?

Accepted Answer

$$\$$int_0$$^2 \$$int_0$$^3 \$$int_0$$^{12-2x-3y} dz \, dy \, dx$$

Question 3

What is the geometric interpretation of the triple integral $$\int\int\int_E 1 \, dV$$ over a region $$E$$?

Accepted Answer

It gives the volume of $$E.$$

Question 4

Suppose you want to find the average value of a function $$f(x, y, z)$$ over a region $$E$$ with volume $$V. $$Which formula should you use?

Accepted Answer

$$\dfrac{1}{V} \int\int\int_E f(x, y, z) \, dV$$

Question 5

Which of the following is NOT a correct description of the order of integration in a triple integral?

Accepted Answer

$$\int\int\int dx \, dx \, dz$$

Question 6

If $$E is $$the region in the first octant bounded by $$x=0$$, $$y=0$$, $$z=0$$, and $$2x+3y+z=12$$, what are the limits for $$z in $$the triple integral for volume?

Accepted Answer

$$0 \leq z \leq 12 - 2x - 3y$$

Question 7

Which of the following best describes the process of evaluating a triple integral over a rectangular box $$[a,b] 	imes [c,d] 	imes [e,f]$$?

Accepted Answer

Integrate with respect to $$x$$, then $$y$$, then $$z$$, using the given limits for each variable.

Question 8

In the context of triple integrals, what does the notation $$\int\int\int_E f(x, y, z) \, dV$$ represent?

Accepted Answer

A sum of the values of $$f(x, y, z)$$ over the volume of $$E.$$

Question 9

Which of the following is a correct step in setting up a triple integral to find the volume of a region bounded by a plane and the coordinate axes in the first octant?

Accepted Answer

Express the upper limit for $$z as a $$function of $$x$$ and $$y$$ from the plane equation.

Question 10

A chemical reactor has a region $$E in $$the first octant bounded by $$x=0$$, $$y=0$$, $$z=0$$, and $$2x+3y+z=12. If $$the density at any point is $$\rho(x, y, z) = x + y + z$$, which integral gives the total mass?

Accepted Answer

$$\$$int_0$$^2 \$$int_0$$^3 \$$int_0$$^{12-2x-3y} (x + y + z) \, dz \, dy \, dx$$

Question 11

If you want to change the order of integration in a triple integral, what must you do?

Accepted Answer

Adjust the limits of integration to match the new order.

Question 12

Which of the following is a correct interpretation of $$\int\int\int_E f(x, y, z) \, dV in a $$real-world context?

Accepted Answer

It gives the total value of $$f(x, y, z)$$ distributed throughout the volume $$E.$$

Question 13

A medical imaging scan models a tumor as the region $$E in $$the first octant bounded by $$x=0$$, $$y=0$$, $$z=0$$, and $$2x+3y+z=12. If $$the density of the tumor is constant, which integral gives its volume?

Accepted Answer

$$\$$int_0$$^2 \$$int_0$$^3 \$$int_0$$^{12-2x-3y} dz \, dy \, dx$$

Question 14

Which of the following is a valid reason for changing the order of integration in a triple integral?

Accepted Answer

To simplify the computation by making the limits easier to evaluate.

Question 15

If $$f(x, y, z) = 1$$ and $$E is a $$region in space, what does $$\int\int\int_E f(x, y, z) \, dV$$ compute?

Accepted Answer

The volume of $$E.$$

Triple Integrals and Volume Calculation in Calculus

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