3) Triple Integrals

Question 1

Regions in double Integrals

Answer 1

• functions of x and constants for y
• functions of y and constants for x
• Disjoint union of finitely many with no overlaps but common boundaries

Question 2

Area by double integral

Answer 2

Area(R) = double_integral(.dx.dy).

Over R

E.g. Height =1

Question 3

Volume of a solid region between plane and surface in 3 dimensions

Answer 3

If φ(x,y) is non-negative for all points of R then integral is the volume of the solid region between (x,y)-plane and surface z=φ(x,y)

Volume= double_integral( φ(x,y) .dx.dy)
Over R

If takes some negative and positive( above and below the xy plane) then the integrand gives the difference Between the volume of the solid region above and that below

Question 4

Change of variables in an integral

E.g. Region

Answer 4

JACOBIAN

Double integral( ϕ(x,y) .dx .dy ) over R

= Double integral ( ϕ(F(u,v)) | Absolute OF JACOBIAN| .du.dv)
Over R’

R’ is (u,v) region

Remember absolute of J

Conditions:

1) (x,y) = F(u,v) is SMOOTH
2) a bijective correspondence between the points of R and R’ for each (x,y) in R there must be exactly one point (u,v) in R’ st (u,v) =(x,y)

3) the JACOBIAN is non zero at every point of R’

Ans the inverse map F-1 must be smooth

Question 5

Volume of a semi ellipsoid

Answer 5

(x^2/ a^2) + (y^2/b^2) + (z^2/ c^2) =1

Positive constants and we need the volume above the plane
Double integral of function z=f(x,y) .dA

Substitution of x = aucosv, y = businv
JACOBIAN = abu

Volume is 2piabc/3

Question 6

Triple Integrals swapping orders

Answer 6

For region R of R^3 defined as:

x in [a,b] etc for constants not functions

The order can be changed if ϕ is smooth.

DONT FORGET JACOBIAN IF SUBSTITUTING
Otherwise order cannot be changed

Can change order of two if y again between the two e.g. Dydzdx to dydxdz

Question 7

Substitution in triple integral

Answer 7

Triple integral ( ϕ(x,y,z) dxdydz) over R in xyz 
= triple integral ( ϕ(F(u,v,w)) |absolute value of jacobian| dudvdw )

Over R’ in uvw
Conditions:

1) (x,y,z) = F(u,v,w) map must be smooth
2) must give a bijective correspondence between points of region R and (u,v,w) of region R’ one point, for each (x,y,z) in R, (u,v,w) in R’ st F(u,v,w) = (x,y,z)
3) JACOBIAN must be non zero for all points in R’

Question 8

Volume in triple Integrals

Answer 8

Volume under a smooth non negative function can use double integrals

Triple Integrals:

Vol(R) = triple integral (1.dV)

Question 9

Volume of a sphere

Volume of a cylinder

Answer 9

Uses spherical polars substitution with JACOBIAN r^2sintheta

Uses cylindrical polars with JACOBIAN r

If split into octants or parts of the shape make sure to use the right angle boundaries that

Question 10

Integrand extras

Answer 10

If a function is point symmetric around the origin then the integrand of a region could be 0 where

If (x,y,z) in R then (-x,-y,-z) in R also meaning triple integral over R of function is 0

Question 11

Average of function over region

Answer 11

Average ofϕ over R
Is

[Triple Integral ( ϕ(x,y,z .dx.dy.dz) over R ]

DIVIDED BY

[triple integral( 1.dx.dy.dz) over R]

I.e. Integral of function over volume of region