Q. 41 Find the volumes of the solids d... [FREE SOLUTION]

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Chapter 13: Q. 41 (page 1015)

Find the volumes of the solids described in Exercises 41鈥�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,

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

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91影视

Find the volumes of the solids described in Exercises 41鈥�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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91影视

Find the volumes of the solids described in Exercises 41鈥�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

Applied Mathematics

Calculus

Statistics

Discrete Mathematics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

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