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Q38E

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Found in: Page 699

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Graph the solid that the lies between the surfaces$${\bf{Z = }}{{\bf{e}}^{{\bf{ - }}{{\bf{x}}^{\bf{2}}}}}{\bf{cos}}\left( {{{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}} \right){\bf{ and Z = 2 - }}{{\bf{x}}^{\bf{2}}}{\bf{ - }}{{\bf{y}}^{\bf{2}}}$$for $$\left| x \right| \le 1,\left| y \right| \le 1$$.Use a compute algebra system to approximate the volume of this solid correct to four decimal places.

Hence the required volume of the solid is.$$3.0271$$

See the step by step solution

Step 1: Maple input command for solid figure

To sketch the solid that lies between the surfaces$$Z = {e^{ - {x^2}}}\cos \left( {{x^2} + {y^2}} \right){\rm{ and }}Z = 2 - {x^2} - {y^2}$$for$$\left| x \right| \le 1,\left| y \right| \le 1$$,use compute algebra system.

Maple input command:

Enter$$Z = \exp \left( { - {x^2}} \right)\cos \left( {{x^2} + {y^2}} \right),z = 2 - {x^2} - {y^2}$$in maple and select plot builds.

Output$$\Rightarrow Z = {e^{ - {x^2}}}\cos \left( {{x^2} + {y^2}} \right){\rm{ and }}Z = 2 - {x^2} - {y^2}$$

Step 2: Forming function for volume

The volume of a solid over a rectangle region can be calculated by the appropriate integral.

$$\int\limits_a^b {\int\limits_c^d {f\left( {x,y} \right)} } dydx$$

To find the volume of the solid, need to obtain the limits of integration and appropriate function of integrates.

Since, the solid, need to obtain the limits of integration and appropriate function to integrates.

Since, the solid lies in between$${e^{ - {x^2}}}\cos \left( {{x^2} + {y^2}} \right) \le z \le 2 - - {x^2} - {y^2}$$, our function is,

$$f\left( {x,y} \right) = \left( {2 - {x^2} - {y^2}} \right) - \left( {{e^{ - {x^2}}}\cos \left( {{x^2} + {y^2}} \right)} \right)$$

Step 3: Determining limits

From the problem, the solid is bounded by$$- 1 \le x \le 1{\rm{ and }} - 1 \le y \le 1{\rm{ or }}\left| x \right| \le 1{\rm{ and }}\left| y \right| \le 1$$. This allows us to write the volume of the solid as,

$$V = \int\limits_0^1 {\int\limits_0^1 {\left( {2 - {x^2} - {y^2}} \right)} } - \left( {{e^{ - {x^2}}}\cos \left( {{x^2} + {y^2}} \right)} \right)dydx$$

Now, use maple software to solve the integrated$$V$$.

Step 4: Maple software command for volume

$${\mathop{\rm int}} \left( {{\mathop{\rm int}} \left( {2 - {x^2} - {y^2} - \exp \left( { - {x^2}} \right)\cos \left( {{x^2} + {y^2}} \right)} \right)} \right)$$,

$$y = - 1.1,x = - 1..1$$, then press enter.

Then exact value output would be,$$3.027069072 - 0.1$$.

Rounded to four decimal is$$V = 3.0271$$

Hence the required volume of the solid is.$$3.0271$$