Answered: Find the surface area of the ellipse…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the surface area of the ellipse that lies on the plane z=11-3x+2y and inside the cylinder x^2+y^2=9

Expert Solution

Step 1

The plane $z = 11 - 3 x + 2 y$ and cylinder is $x^{2} + y^{2} = 9$ .

Obtain the partial derivatives of $z = 11 - 3 x + 2 y$ :

$\frac{\partial z}{\partial x} = - 3 \frac{\partial z}{\partial y} = 2$

Step 2

Substitute $\frac{\partial z}{\partial x} = - 3, \frac{\partial z}{\partial y} = 2$ in $A (S) = \begin{array}{rcl} \iint_{D} \end{array} \sqrt{1 + {(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2}} d A$ :

$\begin{array}{rcl} A (S) & = & \iint_{D} \sqrt{1 + {(- 3)}^{2} + {(2)}^{2}} d A \\ = & \iint_{D} \sqrt{1 + 9 + 4} d A \\ = & \sqrt{14} \iint_{D} d A \end{array}$

Note that, D is the projection of the xy plane.

Also it is required to find surface area of the ellipse inside cylinder is $x^{2} + y^{2} = 9$ .

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