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9. (10 points) Evaluate the line integral

y³dx – x³dy, where C is the circle x² + y² = 8 with positive orientation.

其他申論題

5. (10 points) Find the volume of the solid of revolution about the x-axis bounded by the curvef(x) =with x ∈ [1,0∞).

6. (10 points) Consider the power series Find its maximal domain of convergence.

7. (10 points) Recall that the hyperbolic tangent function tanh x=,x∈Ris one to one and has the inverse function f(x) = tanh-1x. Find the Maclaurin series off(x) = tanh-1x.

8. (10 points) Find the absolute maximum of f(x, y) = e-xy on the region x² + 4y² ≤ 1.

10. (10 points) Let F(x, y, z) = xi + yj – 2zk. Suppose that the closed surface S is defined by the paraboloid y = x² + z² with 0 ≤ y ≤ 1. Find the flux of F outward across S.

(一) 假設兩流體為逆流，總熱傳係數(Overall Heat Transfer Coefficient) U = 350 W/m2∙K，試計算水的出口溫度。(5 分)

(二) 兩流體為逆流狀態，試計算熱傳面積。(5 分)

(三) 若兩流體為順流狀態，試計算熱傳面積。(10 分)

(一) 在操作精餾塔時，若提高塔頂回流比，其產生之優缺點為何？(6 分)

(二) 進行氣體之吸收或萃取單元操作時，流體以順流式接觸相較於逆流式接觸之優缺點為何？(4 分)

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