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Z
0
1

f(x; y)dydx
(a) 画出 xy 平面中的积分区域。
(b) 求积分阶相反的等效二重积分。
(c) 如果给定密度（单位面积质量），则确定积分区域的质量

px2 + y2 和下界

p9 x2 y2。
(a) 草图 E。
(b) 使用圆柱坐标计算 E 的体积。
(c) 使用球坐标计算 E 的体积。

F(x; y; z) = e2x+1
i + (4z y(y + 8) x2
)j + 3xk
(a) 判断 F 是否保守。
(b) 使用斯托克斯定理来评估功积分
Z
C

(c) 根据增加的参数 t 找到 C 的参数化。

Question 1 (7 marks)
Consider the double integral
Z 
0
Z 1
cos x
f(x; y)dydx
(a) Sketch the region of integration in the xy-plane.
(b) Obtain an equivalent double integral with the order of integration reversed.
(c) Determine the mass of the region of integration if the density (mass per unit area) is given
by f(x; y) = x.
Question 2 (12 marks)
Let E be the solid region in R3 bounded above by the cone z =
px2 + y2 and bounded below
by the hemisphere z =
p9 x2 y2.
(a) Sketch E.
(b) Calculate the volume of E using cylindrical coordinates.
(c) Calculate the volume of E using spherical coordinates.

Question 3 (13 marks)
Let S be the region on the surface z = 2y + 1 satisfying x2 + y2  4 oriented with upward unit
normal, let C be the closed curve on z = 2y + 1 bounding S, and let F be the vector eld
F(x; y; z) = e2x+1
i + (4z y(y + 8) x2
)j + 3xk
(a) Determine whether or not F is conservative.
(b) Use Stokes’ theorem to evaluate the work integral
Z
C
F  dr
(c) Find a parametrisation for C in terms of an increasing parameter t.

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