In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables :
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=9x2y2 and the xyplane. 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 ex2dx 9.
Find the surface area of the part of the surface z=9x2 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 emailed to you at the address you provide. MCQ Quiz: Calculus of Multiple Variables Part IIIGoogle Spreadsheet Form
MCQ Quiz #2
Companion MCQ Quiz #2 for this topic test how much you know about the topic. Your score will be emailed to you at the address you provide.
MCQ Quiz Multivariate Calculus III  Quiz 2Google Spreadsheet Form In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables :
