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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): MCQ Quiz #1Companion 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 #2Companion 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. In case you'd like to take a look at other tutorials we have, related to Calculus of multiple variables : |
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