Chapter 5: Q47P (page 249)

Find the volume in the first octant bounded by the coordinate planes and the plane $x + 2 y + z = 4$ .

Short Answer

Expert verified

The volume in the first octant is $\frac{16}{3} .$

Step by step solution

Definition of double integral and mass formula

Double integralof $f(x,y)$ over the area in the plane $(x,y)$ as the limit of this sum, and write it as $\iint_{A} f(x,y)dxdy.$

Drawing the area bounded by the curve in xy plane

The area is bounded by region for the integration in plane.

Calculation of the volume under the curve

In this case, the plane $x + 2 y + z = 4$ intersect at $(4,0,0)$ , $(0,2,0)$ and $(0,0,4)$ points.

Calculation of the volume integral:

$V = \int_{0}^{4} d x \int_{0}^{2 - x / 2} d y \int_{0}^{4 - x - 2 y} d z = \int_{0}^{4} d x \int_{0}^{2 - x / 2} (4 - x - 2 y) d y = {\int_{0}^{4} d x (4 y - x y - y^{2})|}_{0}^{2 - x / 2} = \int_{0}^{4} [4 (2 - \frac{x}{2}) - x (2 - \frac{x}{2}) - {(2 - \frac{x}{2})}^{2} - 0] d x = \int_{0}^{4} [4 (2 - \frac{x}{2}) - x (2 - \frac{x}{2}) - {(2 - \frac{x}{2})}^{2}] d x$

Calculation further to obtain the final value.

$\int_{0}^{4} [4 (2 - \frac{x}{2}) - x (2 - \frac{x}{2}) - {(2 - \frac{x}{2})}^{2}] d x = \int_{0}^{4} (\frac{x^{2}}{4} - 2 x + 4) d x = {[\frac{x^{3}}{12} - x^{2} + 4 x]|}_{0}^{4} = \frac{16}{3}$

Therefore, the value is $\frac{16}{3}$

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Find the volume in the first octant bounded by the coordinate planes and the plane $x + 2 y + z = 4$ .

Short Answer

Step by step solution

Definition of double integral and mass formula

Drawing the area bounded by the curve in xy plane

Calculation of the volume under the curve

One App. One Place for Learning.

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Find the volume in the first octant bounded by the coordinate planes and the plane x+ 2y+z= 4.

Short Answer

Step by step solution

Definition of double integral and mass formula

Drawing the area bounded by the curve in xy plane

Calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Physics of Motion

Work Energy and Power

Force

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Particle Model of Matter

Study anywhere. Anytime. Across all devices.

Find the volume in the first octant bounded by the coordinate planes and the plane $x + 2 y + z = 4$ .