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Calculus of Multiple Variables - Tutorial with Problems, Solutions, MCQ Quizzes - Part III : Multiple Integrals, Double and Triple Integrals




Here's the outline of what we'll cover in this tutorial

Multiple Integrals

1. Multiple Integrals : Extensions of single integral to two , three or more dimensions is called Multiple integrals.
2. Area of a region : Computing the area of a given region using double Integrals.
3. Volume under a surface : Computing volume of  the region using multiple integrals.
4. Computing the average using multiple integrals.
5. Computing the center of gravity of a solid given the density as a function of (x,y,z).
6. Computing the moment of inertia using double integral.

Using Triple Integrals to compute
Volume of a region, center of gravity and moment of inertia of a solid.

Making transformations to help us change the variables in double and triple integrals.


1.  Find the volume of the solid under the plane z = 2x + 4y and over
the rectangle [3, 12] × [2, 4].
2.    Find the area of the triangle
R = {(x, y) :0xb,0ymx }
using double integrals and show that it gives the usual formula y = (base)(height)/2
3 . Find the volume of the tetrahedron bounded by the coordinate planes
and the plane z = 6 − 2x − 3y.
4 . Find the volume of the solid in the first octant bounded by the
paraboloid   z=9-x2-y2   and the xy-plane.
5. Solve the above problem using  cylindrical coordinates
6. Find a formula for the area of a circle of radius a using double   integrals.
7. Find a formula for the area of a circle of radius a using polar  integrals.
8. Find the value of -e-x2dx
9. Find the surface area of the part of the surface z=9-x2   that lies above the quarter of the circle x2+y2=9 in the first quadrant .
10 .Find a formula for the surface area of a sphere of radius a.
Find the volume of the solid in the first octant bounded by the
surfaces y2+64z2=4 and y=x
13. Find the volume of ball of radius a centered at the origin.
16. Find the Cartesian coordinates corresponding to the point (3,-7/6)
18.  A solid fills the region between two concentric spheres of radii a and b , 0<a<b.
The density at each point is inversely proportional to its square of distance from the region . Find the total  mass.
19. Find the volume of the solid enclosed between the surfaces x2+y2=a2  and  x2+z2=a2
22. The cylinder x2+z2=1  is cut   by the planes  y=0 , z=0  and  x=y.  Find the
volume of the region in the  first octant.
26.  Find the centre of gravity of a plate whose  density  (x,y) is constant  and is bounded by the curves   y=x2  and   y=x+2 . 


Complete Tutorial (MCQ Quizzes after this):



MCQ Quiz #1


Companion MCQ Quiz for this topic- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.


MCQ Quiz: Calculus of Multiple Variables- Part III




MCQ Quiz #2


Companion MCQ Quiz #2 for this topic- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.



MCQ Quiz- Multivariate Calculus III - Quiz 2



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