課程名稱︰微積分甲下
課程性質︰系內必帶
課程教師︰陳宏
開課學院：工學院
開課系所︰土木工程學系
考試日期（年月日）︰2010/6/10
考試時限（分鐘）：50min
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
Make sure to give sufficient reason in each problem or you will NOT get any
credit for your answer.
1.(25 points) Find
∫∫sin(x+y)cos(2x-y) dA
E
where E is the region enclosed by y=2x-1, y=2x+3, y=-x, and y=-x+1.
Solution. Refer to Q1 of 90(II) midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/cala902_II_final.pdf
2.(25 points) Find the surface area of the torus
r(u,v)=(R+rcosu)cosv i+(R+rcosu)sinv j+rsinu k, 0<=u,v<=2π,
where r, R are two constants and r<R.
Solution. Refer to Q2 of 92 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/952Cala2Final_sol.pdf
3.(25 points) Consider the conservative force field
1
F=yz^2 i+(xz^2+ze^(yz)) j+(2xyz+p(y,z)+-----) k,
1+z
where p(y,z) is a continuous differentiable function of y and z and p(0,z)=0.
(a). Determine p(y,z)
(b). Find the potential function of F.
(c). Let C denote the space curve r(t)=ti+t^2j+t^3k, 0<=t<=1.
Evaluate ∫F·dr.
C
Solution. Refer to Q5 og 96 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/962A2Final.pdf
4.(25 points) Evaluate
∫(4y+3x+siny)dx+(3y+2x+xcosy)dy
C
where C is the circle (x-2)^2+(y-5)^2=9.
Solution. Refer to Q5 of 94 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/942cala2_final_so1.pdf
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