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1. L. D. College Of Engineering,
Ahmedabad

2. Calculus
Multiple Integrals
- Triple Integrals

3. Index:-
Triple Integrals
Triple Integrals in Cylindrical Co-
ordinates
Triple Integrals in Spherical Co-
ordinates
Change of order of Integration
Jacobian of several variables

4. Triple Integrals:
The triple integral is defined in a similar manner to that
of the double integral if f(x,y,z) is continuous and
single-valued function of x, y, z over the region R of
space enclosed by the surface S. We sub divide the
region R into rectangular cells by planes parallel to the
three co-ordinate planes(fig 1).The parallelopiped cells
may have the dimensions of δx, δy and δz.We number
the cells inside R as δV1, δV2,…..δVn.

5. .
In each such parallelopiped cell we choose an arbitrary
point in the k th pareallelopiped cell whose volume is
δVk and then we form the sum
=

6. .

7. .

8. .

9. .

10. Triple Integrals In Cylindrical
Coordinates:
 We obtain cylindrical coordinates for space by
combining polar coordinates (r, θ) in the xy-plane with
the usual z-axis.
 This assigns every point in space one or more
coordinates triples of the form (r, θ, z) as shown in
figure.2.

11. .

12. Definition : Cylindrical coordinate
Cylindrical coordinate represent a point P in space by orders triples (r, θ, z) in which
1. (r, θ) are polar coordinates for the vertical projection of P on xy-plane.
2. z is the rectangular vertical coordinates.
The rectangular (x , y , z) and cylindrical coordinates are related by the
usual equations as follow :
x = r cosθ, y = r sinθ , z = z
𝑟2
= 𝑥2
+ 𝑦2
, tanθ =
𝑦
𝑥
.

13. Formula for tripple integral in
cylindrical coordinates
where,volume element in cylindrical coordinates is given
by dV = rdzdrdθ

14. Triple Integrals in Spherical Co-
ordinates:
 Spherical coordinates locate points in space is with two
angles and one distance, as shown in figure.3.
 The first coordinate P = |OP|, is the point’s
distance from the origin.
The second coordinate ф, is the angle OP make with the
positive z-axis.
It is required to lie in the interval 0 ≤ ф ≤ π.
The third coordinate is the angle θ as measured in
cylindrical coordinates.

15. Figure.3

16. Definition : Spherical coordinates
Spherical coordinates represent a point P in ordered triples (ƍ , θ , ф) in
which
1. ƍ is the distance from P to the origin.
2. θ is the angle from cylindrical coordinates.
3. ф is the angle OP makes with the positive z-axis (0 ≤ ф ≤ π).
The rectangular coordinates (x , y, z) and spherical coordinates are related
by the
following equations :
x = ƍ sinф cosθ , y = ƍ sinф sinθ, z = P cosф.

17. Formula for Triple integral in
spherical coordinates:-
where, D = {(ƍ , θ , ф) | a ≤ ƍ ≤ b, α ≤ θ ≤ β,
c ≤ ф ≤ d}
and dV = ƍ2 sin фdƍdф.