Chapter 5: Q49P (page 249)

Find the volume in the first octant bounded by the paraboloid $z = 1 - x^{2} - y^{2}$ , the plane $x + y = 1$ , and all three coordinate planes.

Short Answer

Expert verified

The volume in the first octant is $\frac{1}{3} .$

Step by step solution

Definition of double integral and mass formula

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

Drawing the volume in the first octant bounded by the curve

The volume in the first octant is bounded by the paraboloid $z = 1 - x^{2} - y^{2}$ , the plane $x + y = 1$ , and all three coordinate planes.

Calculation of the volume under the curve

Calculation integral of the volume.

$V = \int_{0}^{1} d x \int_{0}^{1 - x} d y \int_{0}^{1 - x^{2} - y^{2}} d z = \int_{0}^{1} d x \int_{0}^{1 - x} d y (1 - x^{2} - y^{2}) = {\int_{0}^{1} d x (y - x^{2} y - \frac{1}{3} y^{3})|}_{0}^{1 - x} = \int_{0}^{1} ((1 - x) - x^{2} (1 - x) - \frac{1}{3} {(1 - x)}^{3}) d x$

Substitute, $u = 1 - x$ , then $d u = - d x$ .

Further calculation of the volume under the curve

The volume is given by,

$V = - \int_{1}^{0} d u (u - {(1 - u)}^{2} u - \frac{1}{3} u^{3}) = \int_{0}^{1} (- \frac{4}{3} u^{3} + 2 u^{2}) d u = {[- \frac{1}{3} u^{4} + \frac{2}{3} u^{3}]|}_{0}^{1} = \frac{1}{3}$

Therefore, the value is $\frac{1}{3} .$ ,

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Find the volume in the first octant bounded by the paraboloid $z = 1 - x^{2} - y^{2}$ , the plane $x + y = 1$ , and all three coordinate planes.

Short Answer

Step by step solution

Definition of double integral and mass formula

Drawing the volume in the first octant bounded by the curve

Calculation of the volume under the curve

Further calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

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Find the volume in the first octant bounded by the paraboloid z=1-x2-y2, the plane x+y=1, and all three coordinate planes.

Short Answer

Step by step solution

Definition of double integral and mass formula

Drawing the volume in the first octant bounded by the curve

Calculation of the volume under the curve

Further calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Measurements

Classical Mechanics

Force

Work Energy and Power

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

Find the volume in the first octant bounded by the paraboloid $z = 1 - x^{2} - y^{2}$ , the plane $x + y = 1$ , and all three coordinate planes.