INTEGRAL CALCULUS – TRIPLE INTEGRATION IIT JAM NOTES

INTEGRAL CALCULUS NOTES WITH PRACTICE QUESTIONS – TRIPLE INTEGRATION

Complete set of notes on Integral Calculus that is exclusively compiled & managed by Dr Gajendra Purohit. To learn the concept in detail – watch. The notes and Solved Examples for IIT JAM Mathematics have been prepared according to the Mathematics exam syllabus.

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TRIPLE INTEGRATION

WORKING RULE :

• Consider a function u = f(x,y,z), then

•  If all the limit are constant then

Example: 

Solution –

 

 

 

Example: 

Solution – 

S is the sphere x² + y² + z² = 1.

Since x, y, z are all positive we have to consider only the positive octant of the sphere.

Now x² + y² + z² = 1 so that z = √(1 – x² – y²)

The projection of the sphere on xy plane is the circle x² + y² = 1.

This circle is covered as y-varies from 0 to √(1 – x²) and x varies from 0 to 1.

 

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Answer in comment section –

Question 1) 

Question 2)

   

Question 3)

Question 4)

        

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VOLUME BY TRIPLE INTEGRATION 

Suppose V is the volume of the solid region D,then

Example: The volume of the solid which is bounded by the cylinder x² + y² = 1 and plane y + z = 1 and z = 0 is 

Solution –

We know that

where x = r cos θ, y = r sin θ, dxdy = rdrdθ

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Answer in comment section –

Question 1)  The volume of tetrahedron bounded by the plane z = 0, x = 0, y = 0 & y + z – x = 1 ?

Question 2) The volume of the closed region bounded by the surface x² + y² = 2x, z = -1 and z = 1 is ?     

Question 3) Find the volume of the region in the first octant bounded by the surfaces x = 0, y = x, y = 2 – x², z = 0 and z = x² ?

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