Chapter 13: Q. 41 (page 1015)

Find the volumes of the solids described in Exercises 41–44. The portion of the first octant bounded by the coordinate planes and the plane $3 x + 4 y + 6 z = 12$

Short Answer

Expert verified

The volume of the described solid is: 4 units

Step by step solution

Step 1. Given Information

There is a portion of the first octant bounded by the coordinate planes and the plane,

$3 x + 4 y + 6 z = 12$

Step 2. Finding the volume of given portion of octant

The double integral to find the volume of portion of octant is given by,

$\int_{0}^{4} \int_{0}^{\frac{12 - 3 x}{4}} \frac{(12 - 3 x - 4 y)}{6} d y d x = \int_{0}^{4} {(\frac{12 y}{6} - \frac{3 x y}{6} - \frac{4 y^{2}}{12})}_{0}^{\frac{12 - 3 x}{4}} d x = \int_{0}^{4} {(2 y - \frac{x y}{2} - \frac{y^{2}}{3})}_{0}^{\frac{12 - 3 x}{4}} d x = \int_{0}^{4} (\frac{2 (12 - 3 x)}{4} - \frac{x (12 - 3 x)}{8} - \frac{{(12 - 3 x)}^{2}}{48}) d x = \int_{0}^{4} (6 - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3 x^{2}}{8} - \frac{3 (16 + x^{2} - 8 x)}{16}) d x = \int_{0}^{4} (6 - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3 x^{2}}{8} - 3 - \frac{3 x^{2}}{16} + \frac{3 x}{2}) d x = \int_{0}^{4} (3 - \frac{3 x}{2} + \frac{3 x^{2}}{16}) d x = {(3 x - \frac{3 x^{2}}{4} + \frac{x^{3}}{16})}_{0}^{4} = (3 (4 - 0) - \frac{3 (4^{2} - 0^{2})}{4} + \frac{(4^{3} - 0^{3})}{16}) = 12 - 12 + 4 = 4$

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Find the volumes of the solids described in Exercises 41–44. The portion of the first octant bounded by the coordinate planes and the plane $3 x + 4 y + 6 z = 12$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the volume of given portion of octant

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Find the volumes of the solids described in Exercises 41–44. The portion of the first octant bounded by the coordinate planes and the plane3x+4y+6z=12

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the volume of given portion of octant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Statistics

Applied Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the volumes of the solids described in Exercises 41–44. The portion of the first octant bounded by the coordinate planes and the plane $3 x + 4 y + 6 z = 12$