Double and triple integration, Study notes of Engineering Mathematics

Download Double and triple integration and more Study notes Engineering Mathematics in PDF only on Docsity!

WEEK 6

1 Double Integrals 1.2 Polar Coordinates 2 Triple Integrals Multivariable functions 2.1 Cartesian Coordinates

1.2 in Polar Coordinates Useful for a region R in circular type shape Recall: Polar coordinates of a point P is written as. r - the distance of P from the pole

• the angle measured from the polar axis to the line OP (radial axis). , , It can be shown that

( r ,q)

q

x = r cos q y = r sin q

2 2 2 x + y = r 2 1 ( ) ( ) ( , ) ( cos , sin ) r R r f x y dA f r r rdrd b^ q a q

= q q q

ò ò ò ò

dA = rdrd q

Limit for q must be constant (fixed)

(Radial axis) (Polar axis) O r q P ( r , q) x y

Let’s practice determining limits (R is provided)

Example 1: Evaluate the following integrals by changing to polar coordinates: 2 4 2 2 2 0 0 cos( ) x x y dydx

ò ò

Example 3 :

Example 4 : Evaluate the following integrals by changing to polar coordinates: (^1 ) 2 2 0 0 ( ) y y dxdy x + y ò ò

Take note on common polar equations:

Example 6 :

Example 8 :

Example 9 :

¨ dV could be in one of the following order ¨ For a rectangular solid (cuboid), G : , = = = We can easily swap the order of integration and limit around (because all the limits are constant). BUT !– for Nonrectangular solid G , you cannot simply swap the order of integration and limit around. You have to refer to the sketched solid G and see how it changes. dz dy dx dz dx dy dy dx dz dy dz dx dx dy dz dx dz dy a £ x £ b , c £ y £ d , k £ z £ l ò ò ò G

f ( x , y , z ) dV ( , , )

l d b k c a f x y z dxdydz ò ò ò ( , , ) l b d k a c f x y z dydxdz ò ò ò

l d b k c a f x y z dxdydz ò ò ò

How to determine the limit?? Sketch the solid G Choose order of integration (choose where you want your solid to be projected to) Determine the limit:

• R limit - recall double integrals (Cartesian)
• third limit - sketch stripe in x/y/z direction accordingly 1 2 3

Example 1 1 : If G is the solid region in the first octant bounded by , , xy - plane and yz - plane, evaluate . 2 y = x z +^ y =^1

G

z dV

ò ò ò

Example 1 2 :