Chapter 13: Q.26 (page 1082)

Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{1}^{5} \int_{- 2}^{0} \int_{- 1}^{3} d z d y d x$

Short Answer

Expert verified

The obtained integral is 32.

Step by step solution

Step 1. GIven information.

Consider the given information.

$\int_{1}^{5} \int_{- 2}^{0} \int_{- 1}^{3} d z d y d x$

Step 2. Graph the solid.

By using the limits, sketch the given solid.

$\begin{array}{l} - 1 \leq z \leq 3 \\ - 2 \leq y \leq 0 \\ 1 \leq x \leq 5 \end{array}$

And,

Step 3. Calculate the integral.

Integrate by using limit and solve.

$\begin{array}{rcl} \int_{1}^{5} \int_{- 2}^{0} \int_{- 1}^{3} d z d y d x & = & \int_{1}^{5} \int_{- 2}^{0} {[z]}_{- 1}^{3} d y d x \\ = & \int_{1}^{5} \int_{- 2}^{0} {[3 - (- 1)]}_{- 1}^{3} d y d x \\ = & 4 \int_{1}^{5} {[y]}_{- 2}^{0} d x \\ = & 4 \int_{1}^{5} [0 - (- 2)] d x \\ = & 8 {[x]}_{1}^{5} \\ = & 8 [5 - 1] \\ = & 32 \end{array}$

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Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{1}^{5} \int_{- 2}^{0} \int_{- 1}^{3} d z d y d x$

Short Answer

Step by step solution

Step 1. GIven information.

Step 2. Graph the solid.

Step 3. Calculate the integral.

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Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.∫15∫-20∫-13dzdydx

Short Answer

Step by step solution

Step 1. GIven information.

Step 2. Graph the solid.

Step 3. Calculate the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Evaluating triple integrals: Each of the triple integrals that follow represents the volume of a solid. Sketch the solid and evaluate the integral.
$\int_{1}^{5} \int_{- 2}^{0} \int_{- 1}^{3} d z d y d x$