L. D. College Of Engineering, Ahmedabad

CalculusMultiple Integrals

- Triple Integrals

Index:-Triple IntegralsTriple Integrals in Cylindrical Co-

ordinatesTriple Integrals in Spherical Co-

ordinatesChange of order of IntegrationJacobian of several variables

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.

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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 =

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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.

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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 = + , tanθ =

Formula for tripple integral in cylindrical coordinates

where,volume element in cylindrical coordinates is given by dV = rdzdrd

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’sdistance 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.

Figure.3

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.

Formula for Triple integral in spherical coordinates:-

where, D = {(ƍ , θ , ф) | a ≤ ƍ ≤ b, α ≤ θ ≤ β, c ≤ ф ≤ d} and dV = dƍdф.