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**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):**

Multivariable Calculus Tutorial 3

**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

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**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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