Question

Enclosed by \(z = 1 - {x^2} - {y^2}\) and \(z = 0\)

Short answer

We should know about computer algebra system to find the exact volume of the solid.

Step-by-step solution

Given data

Consider \(z = 1 - {x^2} - {y^2}\) and \(z = 0\)

Need to find the exact volume of the solid using computer algebra system.

Recall that the volume of a solid over a region can be calculated by the appropriate integral .

Note that \(z = 1 - {x^2} - {y^2}\) and \(z = 0\)intersect in the circle \({x^2} + {y^2} = 1\) (or) \(R = \left\{ {(x,y)| - 1 \le x \le 1, - \sqrt {1 - {x^2}} \le y \le \sqrt {1 - {x^2}} } \right\}\).

To find the volume of the solid, we need to obtain the limits of integration and appropriate function to integrate. On R we have that \(0 \le z \le 1 - {x^2} + {y^2}\). So, our function is, \(f(x,y) = 1 - {x^2} + {y^2}\).

From R we have \( - 1 \le x \le 1\) and \( - \sqrt {1 - {x^2}} \le y \le \sqrt {1 - {x^2}} \),\(0 \le y \le 4\). This allows us to write the volume of the solid as \(\int\limits_{ - 1}^1 {\int\limits_{ - \sqrt {1 - {x^2}} }^{\sqrt {1 - {x^2}} } {(1 - {x^2} + {y^2})} dydx} \).

Maple software to evaluate integral

Next we use maple software to evaluate integral.

\(\int\limits_{ - 1}^1 {\int\limits_{ - \sqrt {1 - {x^2}} }^{\sqrt {1 - {x^2}} } {(1 - {x^2} + {y^2})} dydx} \)

Using the symbolic command, which is shown below:

\(\begin{aligned}{l}{\mathop{\rm int}} \left( {{\mathop{\rm int}} \left( {1 - x\^2 - y\^2,y = - sqrt(1 - x\^2).sqrt(1 - x\^2)} \right),x = - 1..1} \right);\\ > {\mathop{\rm int}} \left( {{\mathop{\rm int}} \left( {1 - {x^2} - {y^2},y = - sqrt(1 - {x^2}).sqrt(1 - {x^2})} \right),x = - 1..1} \right);\\ = \frac{1}{2}\pi \end{aligned}\)

This output the volume of \(\frac{\pi }{2}\).

Hence the volume of the solid is \(\frac{\pi }{2}\).