Chapter 13: Q.25 (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.
$\int_{0}^{2} \int_{1}^{4} \int_{- 2}^{3} d x d y d z$

Short Answer

Expert verified

The value of the triple integral is 30.

Step by step solution

Step 1. Given information.

Consider the given information.

$\int_{0}^{2} \int_{1}^{4} \int_{- 2}^{3} d x d y d z$

Step 2. Graph the solid.

By using the limits, sketch the given solid.

$\begin{array}{l} - 2 \leq x \leq 3 \\ 1 \leq y \leq 4 \\ 0 \leq z \leq 2 \end{array}$

And,

Step 3. Calculate the integral.

Integrate by using limit and solve.

$\begin{array}{rcl} \int_{0}^{2} \int_{1}^{4} \int_{- 2}^{3} d x d y d z & = & \int_{0}^{2} \int_{1}^{4} {[x]}_{- 2}^{3} d y d z \\ = & \int_{0}^{2} \int_{1}^{4} [3 - (- 2)] d y d z \\ = & 5 \int_{0}^{2} {[y]}_{1}^{4} d z \\ = & 5 \int_{0}^{2} [4 - 1] d z \\ = & 15 {[z]}_{0}^{2} \\ = & 30 \end{array}$

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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.
$\int_{0}^{2} \int_{1}^{4} \int_{- 2}^{3} d x d y d z$

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.∫02∫14∫-23dxdydz

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

Statistics

Theoretical and Mathematical Physics

Calculus

Decision Maths

Discrete Mathematics

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.
$\int_{0}^{2} \int_{1}^{4} \int_{- 2}^{3} d x d y d z$