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Q38E

Expert-verifiedFound in: Page 699

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.\(3.0271\)

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

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,

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

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.\(3.0271\)

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