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
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.
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 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 e-mailed 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 :
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