課程名稱︰微積分甲下
課程性質︰土木系大一必修
課程教師︰洪立昌
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015年5月19日
考試時限（分鐘）：60分鐘
試題 :
Instructions : (130 points) Show all details as possible as you can.
(25pts)
(15.4 Double Integrals in Polar Coordinates)
(√3)/2 √(1-y^2)
1.Please evaluate ∫ ∫ sin(x^2 + y^2)dxdy by answering the following
0 y/√3
questions in order
(i) Find the intersection point(s) of x = √(1-y^2) and x = y/√3 in the
first quadrant.(3 %)
(ii) Find the angle (less then π/2) between the line x = y/√3 and the x-axis
(3 %)
(iii) Plot the region of the integral domain; label the intersection point(s)
in (i) and the angle in (ii). (3 %)
(iv) Describe the integral domain in polar coordinates. (3 %)
√3 /2 √(1-y^2)
(v) Evaluate ∫ ∫ sin(x^2 + y^2) dxdy using the results in
0 y/√3
(iii) and (iv). (13%)
(25pts)
(15.6 Surface Area)
2. Find the surface area of the part (denote by A) of the sphere x^2 + y^2 +
z^2 = 4 that is above the plane z = 1 by answering the following questions
in order :
(i) Describe A by z = f(x,y), where x,y ∈ D. Find f(x,y) and D.(8%)
(ii) Write down the integral formula of the area of A in rectangular
coordinates. (no need to evaluate it here) (5%)
(iii) Write down the integral formula of the area of A in polar coordinates.
Then find the surface area of A. (12%)
(25pts)
(15.9 Triple Integrals in Spherical Coordinates)
3. Find the volume of the solid S that lies above the cone z = √(x^2 + y^2)
and below the sphere x^2 + y^2 + z^2 = 1 by answering the following
questions in order :
(i) Describe z = √(x^2 + y^2) in spherical coordinates x = ρsinψcosθ,
y = ρsinψsinθ,z = ρcosθ. (4%)
δ(x , y , z)
(ii) Write down the Jacobian ────── .(no need to proof it.)(5%)
δ(ρ,ψ,θ)
ps.δ代表偏微分符號 ( ptt打不出來 QQ )
(iii) Find the volume of the solid S using the results in (i) and (ii). (16%)
(25pts)
(15.10 Change of Variables in Multiple Integrals)
4. Please elaluate ∫∫ cos ((y-x)/(y+x)) dA, where R is the trapezoidal region
R
with vertices (1,0),(2,0),(0,2),and(0,1) by answering the following question
in order:
δ(x , y)
(i) Let u = y-x , v = y+x and find the Jacobian ─────.(5%)
δ(u , v)
(ii) Plot the region of the integral domain R in the uv-plane and label all
the vertices of it. (5%)
(iii) Evaluate ∫∫ cos((y-x)/(y+x))dA using the results in (i) and (ii) (15%)
R
(30pts)
(15.7 Triple Integrals : Problem Plus and Bonus)
x y z x
5. Assume that f is continuous.Prove ∫∫∫f(t)dtdzdy = 1/2 ∫ (x-t)^2 f(t)dt
0 0 0 0
(1) by answering the following questions in order:
y z
(i) Evaluate ∫∫ f(t)dtdz by changing the order of integration.(10%)
0 0
(ii) Using the result in (i), prove (1) again by changing the order of
integration.(10%)
(iii) If you finish (i) and (ii) , could you use similar arguements to
conjecture and evaluate the following quadruple (consisting of four
partsof members) intrgral? (10%)
s x y z
∫∫∫∫ f(t) dtdzdydx.
0 0 0 0