Josephood7

Integrate [math] over the region in the first octant [math] above the parabolic cylinder [math] and below the paraboloid [math].

Answer : 

Suppose [math] is the solid bounded by the plane [math], the surface [math], and the planes [math] and [math]. Write an iterated integral in the form below to find the volume of the solid [math].

[math]  [math]
with limits of integration

A = 
B = 
C = 
D = 
E = 
F = 

Each triple integral calculates the volume of one of the solids pictured below. Match each integral with the label A - E of its corresponding solid. As always, you may click on the thumbnail image to produce a larger image in a new window. Just take your time; a process of elimination will help with matches that are not obvious.

 1. [math]

 2. [math]

 3. [math]

 4. [math]

 5. [math]

 

 

A	B	C	D	E

 

Note: You can earn partial credit on this problem

Let [math] be the solid half-cone bounded by [math], [math] and the [math]-plane with [math], and let Let [math] be the solid half-cone bounded by [math], [math] and the [math]-plane with [math].

For each of the following, decide (without calculating its value) whether the integral is positive, negative, or zero.

(a) [math] is  ? positive negative zero 

(b) [math] is  ? positive negative zero 

(c) [math] is  ? positive negative zero 

Evaluate the triple integral [math] where [math] is bounded by the parabolic cylinder [math] and the planes [math] and [math].