Chapter 5: Q48P (page 249)

Find the volume in the first octant bounded by the cone $z^{2} = x^{2} - y^{2}$ and the plane $x = 4$ .

Short Answer

Expert verified

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

Step by step solution

Definition of double integral and mass formula

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

Draw the volume in the first octant bounded by the curve

The volume in the first octant bounded by the cone $z^{2} = x^{2} - y^{2}$ and the plane $x = 4$ .

Calculation of the volume under the curve.

Calculation integral of the volume.

$V = \int_{0}^{4} d x \int_{0}^{x} d y \int_{0}^{\sqrt{x^{2} - y^{2}}} d z = \int_{0}^{4} d x \int_{0}^{x} d y (\sqrt{x^{2} - y^{2}})$

Let $y = x \sin θ$ then $d u y = x \cos d θ$ .

The integration bound changed to 0 and $\frac{π}{2}$ because when $x \sin θ = 0$ the value $θ$ will be 0 and when $y = x \sin θ$ the value $θ$ will be $\frac{π}{2}$ .

Further calculation of the volume under the curve

Substitute changed limits to calculate limits.

$V = \int_{0}^{4} d x \int_{0}^{π / 2} x \cos θ d θ \sqrt{x^{2} - x^{2} \sin^{2} θ} = \int_{0}^{4} x^{2} d x \int_{0}^{π / 2} \cos^{2} θ d θ = {\frac{1}{4} x^{3}|}_{0}^{4} \int_{0}^{π / 2} \frac{1 + \cos 2 θ}{2} d θ$

Calculation further to obtain the final value.

$V = {\frac{1}{4} x^{3}|}_{0}^{4} \int_{0}^{π / 2} \frac{1 + \cos 2 θ}{2} d θ = \frac{56}{3} \frac{1}{2} [\frac{π}{2} + {\frac{1}{2} \sin 2 θ|}_{0}^{π / 2}] = \frac{16}{3} π$

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

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Find the volume in the first octant bounded by the cone $z^{2} = x^{2} - y^{2}$ and the plane $x = 4$ .

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw 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.

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Find the volume in the first octant bounded by the conez2=x2-y2 and the plane x=4 .

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw 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

Solid State Physics

Particle Model of Matter

Electrostatics

Physics of Motion

Linear Momentum

Electricity

Study anywhere. Anytime. Across all devices.

Find the volume in the first octant bounded by the cone $z^{2} = x^{2} - y^{2}$ and the plane $x = 4$ .