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題組內容

6. Let R be the region enclosed by y=x, y=x-2, y=1 and y=0.

(a) (10%) Let A be the area of R. A=_____

其他申論題

3. (10%) Compute the indefinite integral =_____

4. (10%) Let Ω be a regioin (on the xy-plane) enclosed by , x=0 and y=e. Let S be the solid obtained by revolving Ω about the y-axis. Let V be the volume of S. V=_____

(a) (10%) If the unit vector u (in the xy-plane) at p is the direction (among all directions at p) along which the height (i.e. the value of z) of S increases most rapidly, then u =_____

(b) (10%) Write H for the plane x+2y+3z=1 and the curve C for the intersection S∩H. Let L be the tangent line to C at P and N be the plane perpendicular to L at P. Then the equation of N is _____

(b) (5%) The double integral =_____

7. (15%) Let Ω be the region {(x, y)|(x-1)2+y2＜1}. Evaluate the double integral

1. xsinxdx=________.

2. For the equation y3-xy2+cos(xy)=2 , at the point (0,1) is ______.

3. Let , where f iscontinuous and g and h are differentiable. Find F'(t)=___________.

4. The maximum value of f(x,y)=y2-x2 on the ellipseis=_________.

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