Chapter 5: Q3P (page 272)

Find the area of the paraboloid $x^{2} + y^{2} = z$ inside the cylinderrole="math" localid="1659151613290" $x^{2} + y^{2} = 9$

Short Answer

Expert verified

The area of the paraboloid $x^{2} + y^{2} = z$ inside the cylinder $x^{2} + y^{2} = 9$ is $A = \frac{π}{6} (37^{3 / 2} - 1)$

Step by step solution

Given Condition

The equation of the paraboloid is $x^{2} + y^{2} = z$

The equation of the cylinder is $x^{2} + y^{2} = 9$

Concept of surface area

An intersection curve consists of the common points of two transversally intersecting surfaces. The surface area of a three-dimensional object is the total area of all its faces.

Draw the diagram

The shaded area is the area of integration.

Calculate the angle

The equation of the paraboloid is $ϕ (x, y, z) = z - x^{2} - y^{2}$

The secant of the angle is found as follows.

$s e c γ = \frac{\sqrt{{|\nabla ϕ|}^{2}}}{|\partial ϕ / \partial z|} = \frac{\sqrt{|- 2 x \hat{x} - 2 y \hat{y} + \hat{z}|}}{1} = \sqrt{4 x^{2} + 4 y^{2} + 1 g}$

Calculate the area.

The area of the surface cut out by the cylinder is calculated as below.

$A = \int_{x} \int_{y} s e c γ d x d y = \int_{0}^{2 x} d θ \int_{0}^{3} \sqrt{4 x^{2} + 4 y^{2} + 1} r d r = \int_{0}^{2 x} d θ \int_{0}^{3} \sqrt{4 r^{2} + 1} d (4 r^{2} + 1) \frac{1}{8} = {\frac{π}{4} \frac{2}{3} {(4 r^{2} + 1)}^{3 / 2}}_{0}^{3} = \frac{π}{6} (37^{3 / 2} - 1)$

Hence, the area of the paraboloid $x^{2} + y^{2} = z$ inside the cylinder $x^{2} + y^{2} = 9$ is $A = \frac{π}{6} (37^{3 / 2} - 1)$ .

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Find the area of the paraboloid $x^{2} + y^{2} = z$ inside the cylinderrole="math" localid="1659151613290" $x^{2} + y^{2} = 9$

Short Answer

Step by step solution

Given Condition

Concept of surface area

Draw the diagram

Calculate the angle

Calculate the area.

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Find the area of the paraboloidx2+y2=zinside the cylinderrole="math" localid="1659151613290" x2+y2=9

Short Answer

Step by step solution

Given Condition

Concept of surface area

Draw the diagram

Calculate the angle

Calculate the area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Wave Optics

Astrophysics

Conservation of Energy and Momentum

Radiation

Quantum Physics

Study anywhere. Anytime. Across all devices.

Find the area of the paraboloid $x^{2} + y^{2} = z$ inside the cylinderrole="math" localid="1659151613290" $x^{2} + y^{2} = 9$