Chapter 12: Q27E (page 708)
Find the volume of the given solid. Enclosed by the cylinders \(z = {x^2},y = {x^2}\) and the planes \(z = 0,y = 4\)
The volume of the given solid can be:
\(V = \frac{{128}}{{15}}\).
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Calculate the iterated integral \(\int\limits_0^1 {\int\limits_0^1 {v{{\left( {u + {v^2}} \right)}^4}{\rm{ }}} } dudv\)
Write the equations in cylindrical coordinates.
a. \({\rm{3x + 2y + z = 6}}\)
b. \({\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{^{\rm{2}}}}{\rm{ = 1}}\)
Evaluate \(\iiint_{\text{E}}{{{\text{x}}^{\text{2}}}}\text{dV}\), where \({\rm{E}}\) is the solid that lies within the cylinder \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 1}}\), above the plane \({\rm{z = 0}}\), and below the cone \({{\rm{z}}^{\rm{2}}}{\rm{ = 4}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{y}}^{\rm{2}}}\).Use cylindrical coordinates.
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.
\(\int\limits_{{\rm{\pi /2}}}^{\rm{\pi }} {\int\limits_{\rm{0}}^{{\rm{2 sin\theta }}} {{\rm{r dr d\theta }}} } \)
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