The solid that lies under the graph of fx y and above

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the solid that lies under the graph of f(x, y) and above the region ~ equals to J times the area of the region; ~ J = 5 and there is point (x, y) E R such that f(x, y) = 6, there is a point in R such that f(x, y) = 4; (c) If Area(R) = 5 and 2 < f(x, y) ::; 10, then f fR f(x, y)dxdy ~ 10. (d) If fmin and fmax are the minimal and the maximal values of f on a region R, then fmin . Area(R) ::; f fR f(x, y)dA :S fmax . Area(R). (3) Which of the following is true: , (a) f JD f(x, y) = J~1 J~1 f(x, y)dydx, where D is the disk of radius 1 centered at ! ~(O,O). "'~:;7~r~-" (b) a lan:in~ lies entirely in the third quadrant, then its moment M x with respect './ . 0 X-axIS IS negative, (c) Center of mass of a lamina is a point which is on the lamina. (d) Given the lamina's mass m and moment M x with respect to z-axis, you can find its moment My with respect to y-axis. (4) Which of the following represents the integral of f(x, y) over the part of the inside of the sphere x 2 + y2 + z2 = 1 that lies in the first octant (x, y, z ~ 0): z 2 ~() \r~ rV 1 - _ y 2 f( )d d ~o Jo x,y,Z xdy Zj 2 0) 000 [ :-1;-£<,/1 (x +1;2) J(x, u. Z dxdydz; ) '"- b rl r~ rV 1 - y 2 ( ) () c Jo Jo Jo f x, y, z dzdydx; {d) [1 rVI-XLy2 cv'f=X1 r( , )d d d . vO :>i1 )0 J $,~, Z 11 z x,
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7 32B MIDTERM 1 Radko (5) ~ch of the following is true about the statements below: (,(sl.?Only statement iii. is correct; (b) ~nts i. and ii. are correct; (c) Only statements i. and iii. correct; (d) All of the statement~ect. Statements: (1 n iterated intergral of a function over the box = {(x,y,z)/x E [O,a], [O,b], [O,e]} 6 " can be expressed in el . rent ways. (ii) One possible iterated integral 0 ver the region D = {(x, y, z)1 0 S; x S; 1, 3x - 3 S; y S; 0, 0 S; z S; 5} is r: t' r 5 Jo Jo Jo f(x, y, z)dzdxdy. (iii) J0 4 ft Jo z J: f(x, y, z) dxdydz = J0 4 .fii J; Jo x f(x, y, z) dydxdz.
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