INTEGRAL CALCULUS – TRIPLE INTEGRATION GATE NOTES

INTEGRAL CALCULUS NOTES WITH PRACTICE QUESTIONS – TRIPLE INTEGRATION

Engineering Mathematics is an important subject in the GATE 2023 Exam, The mathematics in the GATE exam has 13% weight, and it is easy to get all 13% marks with proper GATE Mathematics Study Materials. These 13% marks will make your GATE qualification easy. But you don’t get good study materials for GATE Maths. Here we come for the rescue. We provide one of the best study materials for GATE 2023 mathematics for all branches or papers. 

Complete set of notes on Integral Calculus that is exclusively compiled & managed by Dr Gajendra Purohit. To learn the concept in detail – watch

To Enroll in CSIR NET Mathematics Course – click here

To Enroll in IIT JAM Mathematics Course – click here

To Enroll in GATE Mathematics Course – click here

To Enroll in CUCET (P.G.) Mathematics Course – click here

To Enroll in CSIR NET (General Aptitude) Mathematics Course – click here

For all University Mathematics Course – click here

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.

 

Watch Now – click here

Answer in comment section –

Question 1) 

Question 2)

   

Question 3)

Question 4)

        

To get IIT JAM Previous Year Questions – click here To get structured IIT JAM mathematics notes – click here Give topicwise IIT JAM mathematics quiz – click here

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

Watch Now – click here

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 ?

Thank you for reading, to gain more information about GATE 2023 and learn mathematics and general aptitude from Dr. Gajendra Purohit Sir visit – click here

And for university courses, blogs, quizzes and ask doubts from sir visit – https://www.mathscare.com/