VI. Find the volume bounded by z = 9– x² – y and z = 0. Give a numericalanswer.

Answered: . Find the volume bounded by z = 9 –x²…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.3: Volume And Average Value

Problem 12E

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VI. Find the volume bounded by z = 9– x² – y and z = 0. Give a numerical
answer.

Expert Solution

Step 1

Given that the region is bounded by $z = 9 - x^{2} - y^{2}$ and $z = 0$ .

We know that the volume bounded between $z = f (x, y)$ and $z = g (x, y)$ over the region R is given by $\iint_{R} (f (x, y) - g (x, y)) d x d y$ where R is the projection on to the $x y$ plane.

Here, we have $z = f (x, y)$ as $z = 9 - x^{2} - y^{2}$ and $z = g (x, y)$ as $z = 0$ .

Hence, the volume is given by $V o l u m e = \iint_{R} (9 - x^{2} - y^{2}) d x d y$ where R is the region obtained by solving $z = 9 - x^{2} - y^{2}$ and $z = 0$ .

When $z = 0$ , we get

$z = 9 - x^{2} - y^{2} \Rightarrow 0 = 9 - x^{2} - y^{2} \Rightarrow x^{2} + y^{2} = 9$

Note that, $x^{2} + y^{2} = 9$ represents a circle of radius 3 with center at the origin.

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