Triple Integrals in cylindrical Coordinates FubiniS Theorem If

Triple Integrals in cylindrical Coordinates

Fubini’S Theorem • If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then

How about if we consider another solid, which is NOT rectangular box?

• Not always: – The upper boundary of the solid E is the surface with equation z = u 2(x, y). – The lower boundary is the surface z = u 1(x, y).

In general, we’ve:

• Sometimes it isn’t easy to compute the integral given by

• For instance, is extremely difficult to evaluate.

So what? Changing the coordinate system?

Cylindrical coordinate system

Cylindrical coordinate is just a polar coordinate in space Chapter 7 Notes 3 D Math Primer for Graphics & Game Dev 11

Converting between rectangular and Cylindrical Coordinates Cylindrical to rectangular (r, , z) r Rectangular to Cylindrical r

Evaluating Triple Integrals with Cylindrical Coordinates • We can convert a triple integral from rectangular to cylindrical coordinates by writing x = r cos , y = r sin , leaving z as it is, • using the appropriate limits of integration for z, r, and replacing d. V by ? ? ? ? r dz dr d.

Volume element in cylindrical coordinates: d. V = r dz dr d • The figure shows how. to remember this.

When do we use Cylindrical CS? • Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry.

– For instance, the axis of the circular cylinder with Cartesian equation x 2 + y 2 = c 2 is the z-axis. – In cylindrical coordinates, this cylinder has the very simple equation r = c. – This is the reason for the name “cylindrical” coordinates.

Example-1

Try Me:

Quick survey I feel I understand “Triple Integral with cylindrical coordinates” a) Very well b) With some review, I’ll be good c) Not really d) Not at all

• Evaluate Class (Home) work • where E is the region bounded by the paraboloid y = x 2 + z 2 and the plane y = 4.

Pair-work