Question

1. (i) Find the volume V of the solid bounded by $y = x^2$ and by the planes $y + z = 4$ and $z = 0$ by setting up a triple integral with z as the innermost variable. (ii) Set up a triple integral for finding the volume of the above solid with the innermost variable as $x$. 2. Change the equation from cylindrical to rectangular coordinates and sketch the graphs in the xyz-coordinate system: (i) $z = 4r^2$ and (ii) $r = 4 \sin \theta$. 3. (i) A solid E lies between the paraboloid $z = 24 - x^2 - y^2$ and the cone $z = 2\sqrt{x^2 + y^2}$. Using cylindrical coordinates, find volume of E. (ii) Set up triple integrals to find the centroid of E (i.e., center of mass if the density is a constant).

          1. (i) Find the volume V of the solid bounded by $y = x^2$ and by the
planes $y + z = 4$ and $z = 0$ by setting up a triple integral with z as
the innermost variable.
(ii) Set up a triple integral for finding the volume of the above solid
with the innermost variable as $x$.
2. Change the equation from cylindrical to rectangular coordinates and
sketch the graphs in the xyz-coordinate system: (i) $z = 4r^2$ and (ii)
$r = 4 \sin \theta$.
3. (i) A solid E lies between the paraboloid $z = 24 - x^2 - y^2$ and the
cone $z = 2\sqrt{x^2 + y^2}$. Using cylindrical coordinates, find volume of E.
(ii) Set up triple integrals to find the centroid of E (i.e., center of mass
if the density is a constant).

1. (i) Find the volume V of the solid bounded by y = x^2 and by the
planes y + z = 4 and z = 0 by setting up a triple integral with z as
the innermost variable.
(ii) Set up a triple integral for finding the volume of the above solid
with the innermost variable as x.
2. Change the equation from cylindrical to rectangular coordinates and
sketch the graphs in the xyz-coordinate system: (i) z = 4r^2 and (ii)
r = 4 sinθ.
3. (i) A solid E lies between the paraboloid z = 24 - x^2 - y^2 and the
cone z = 2√(x^2 + y^2). Using cylindrical coordinates, find volume of E.
(ii) Set up triple integrals to find the centroid of E (i.e., center of mass
if the density is a constant).

Added by Connie C.

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