Evaluate triple integral 2(x^3 + xy^2) dV, where E is the solid in the first octant that lies...


(a) Evaluate {eq}\iiint (2x^{3}+xy^{2})dV , {/eq} where {eq}E {/eq} is the solid in the first octant that lies beneath the paraboloid {eq}z=4-x^{2}-y^{2} {/eq}

b) Evaluate the integral, where E is enclosed by the paraboloid {eq}z=5+x^{2}+y^{2} , {/eq}, the cylinder {eq}x^{2}+y^{2}=6 {/eq} and the xy-plane {eq}\iiint e^{z}dV {/eq}

Multiple Integral:

The multiple integral is the type of integral which contains more then one variable. The given multiple integral can be evaluated by using the cylindrical coordinates:

{eq}\displaystyle \iiint_E f(x, y, z) \, dV = \iiint_R f(r, \, \theta, \, z)\cdot r \,\, dz\,dr\,d\theta {/eq}.

Answer and Explanation: 1

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The given integral is:

{eq}\displaystyle \iiint_E 2x^{3} + xy^{2} \, dV {/eq}


E lies beneath the paraboloid {eq}z = 4 - x^{2} -...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples


Chapter 13 / Lesson 10

Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates.

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