Triple Integrals - Engineering Mathematics, Lecture notes of Engineering Mathematics

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MATH

CALCULUS 2

TRIPLE INTEGRALS

Change of Variables Sometimes the evaluation of an iterated integral can be simplified by reversing the order of integration. Illustration: The integral, given above, cannot be evaluated by performing the x-integration first since there is no elementary antiderivative, let and of which is 2xdx. Evaluate this integral by expressing it as an equivalent iterated integral with the order of integration reversed.

Recommended Readings: Calculus Early Transcendentals (Wiley Custom Edition 10ed by Anton)

1. Read pages 1009-
2. Read Example 1 (a) and (b) page 1009
3. Read Example 3, 4, 5, 6, 7, and 8 page 1011- 1014

Exercises 2 Evaluate the integral by first reversing the order of integration.

1. Ans:
2. Ans:

Triple Integral Notations: 3! Possible orders of integration: f(x,y,z)   b a y x y x z x y z x y f x y z dzdydx

2 1 2 1 ( , , )    f e y z y z x y z x y z f x y z dxdydz

2 1 2 1 ( , , )   d c x y x y z x y z x y f x y z dzdxdy

2 1 2 1 ( , , )   b a z x z x y x z y x z f x y z dydzdx

2 1 2 1 ( , , )    f e x z x z y x z y x z f x y z dydxdz

2 1 2 1 ( , , )   d c z y z y x y z x y z f x y z dxdzdy

2 1 2 1 ( , , )

Evaluation of Triple Integration

Example: Evaluate over the rectangular box G defined by the

inequalities

Solution: Of the six possible iterated integrals we might use, we will

choose the first notation,

of integration.

Thus, we will first integrate with respect to z, holding x and y fixed, then

with respect to y, holding x fixed, and finally with respect to x.

b

a

y x

y x

z x y

z x y

f x y z dzdydx

2 1 2 1

Exercises 3 Evaluate the iterated integral

1. Ans: 16

Recommended Readings: Calculus Early Transcendentals (Wiley Custom Edition 10ed by Anton)

1. Read pages 1039 – 1045
2. Read examples 2, 3, 4 and 5 pages 1042 – 1045

Application: Volume of Rectangular Regions by Double Integration https://math.libretexts.org/TextMaps/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/3%3A_Multiple_Integrals/3.1%3A_Double_and_Iterated_Integrals_Over_Rectangles

1. Calculate the volume below the function and above the function and Solution: Using Fubini’s Theorem,

Application: Volume of NonRectangular Regions by Double Integration

2. Find the volume of the solid above and below x = 0 and x = 1. Solution: Sketching the solid, https://math.libretexts.org/TextMaps/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/3%3A_Multiple_Integrals/3.1%3A_Double_and_Iterated_Integrals_Over_Rectangles