# 13 how to find triple integrals use them to find the

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13. How to find triple integrals & use them to find the mass of a solid body with variable density. KEY CONCEPTS AND MAIN DEFINITIONS : 1. Limit of f(x,y) along a specified curve as x, y tends to a, b and in an arbitrary way & continuity of functions of 2 or 3 variables. 2. The gradient of a function and the differentiability of functions of 2 or 3 variables. 3. The total differential of a function & the local linear approximation of a function. 4. The gradient and directional derivatives of a function of 2 or 3 variables. 5. The tangent plane to a surface z = f(x,y) at the point x 0 , y 0, f(x 0 ,y 0 ) . 6. The normal to z = f(x,y) and to a parametric surface r (u,v) = x(u,v), y(u,v), z(u,v) . 7. Local extrema of a function, gradient-zero points & the discriminant . 8. Global extrema (maxima & minima) of a function & critical points . 8. Constrained global extrema of a function and Lagrange multipliers . 10. Double integrals interpreted as the volume under z = f(x,y) and above the region R. 11 Iterated integrals & double integrals over type I & type II regions of the x-y plane. 12. Double integrals in polar coordinates and simple polar regions. 13. Surface area of z = f(x,y) and of a parametric surface r (u,v) = x(u,v), y(u,v), z(u,v) . 14. The triple integral interpreted as the mass of a solid region G with variable density.

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REVIEW FOR TEST #2 - MAIN FORMULAS: FALL 2011 1. f x (x,y) = lim h 0 [f(x+h,y) – f(x,y)] / h ; f y (x,y) = lim k 0 [f(x,y+k) – f(x,y)] / k. f/ x = f x , f/ y = f y , 2 f/ x 2 = f xx , 2 f/ y 2 = f yy , 2 f/ x y = f yx . 2. f (x,y) = f/ x, f/ y ; f (x,y,z) = f/ x, f/ y, f/ z .
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