Chapter 12: Multiple Integrals

The value of the given integral by using polar coordinates.

Find the volume of the given solid. Under the surface \(z = xy\)and above the triangle with vertices \((1,1),(4,1),(1,2)\)

Find the volume of the solid that is enclosed by the cone \({\rm{z = }}\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \)and the sphere \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 2}}\).

A lamina with constant density \(\rho (x,y) = \rho \) occupies the given region. Find the moments of inertia\({I_x}\), \({I_y}\) and the radii of gyration \(\bar \bar x\) and \(\bar \bar y\).

The rectangle.

Question: Use the Midpoint Rule for triple integrals (Exercise \({\rm{22}}\)) to estimate the value of the integral. Divide into eight sub-boxes of equal size. where \({\rm{B = \{ (x,y,z)|0}} \le {\rm{x}} \le {\rm{1,0}} \le {\rm{y}} \le {\rm{1,0}} \le {\rm{z}} \le {\rm{1\} }}\).

Find the volume of the solid that is enclosed by the cone \({\rm{z = }}\sqrt {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \) and the sphere \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 2}}\).Use cylindrical coordinates.

Find the volume of the solid that lies between the paraboloid \({\rm{z = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\)and the sphere \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 2}}\).

Evaluate \(\iint_{\text{E}}{\text{x}}\text{yzdV}\),where E lies between the spheres \({\rm{\rho = 2}}\) and \({\rm{\rho = 4}}\)and above the cone \(\phi {\rm{ = \pi /3}}\).

Calculate the double integral

\(\int {\int\limits_R {\frac{{1 + {x^2}}}{{1 + {y^2}}}dA,R = \{ \left( {x,y} \right)|0 \le x \le - 1,0 \le y \le 1\} } } \)

Question: Use the Midpoint Rule for triple integrals (Exercise) to estimate the value of the integral. Divide into eight sub-boxes of equal size.where \({\rm{B = \{ (x,y,z)|0}} \le {\rm{x}} \le {\rm{4,0}} \le {\rm{y}} \le {\rm{1,0}} \le {\rm{z}} \le {\rm{2\} }}\).