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題組內容

1. Please evaluate the integrals:

(d) ∫∫_Dx + 2y dA, where D is the region bounded by the parabolas y = 2x² and y = 1 + x².(15%)

其他申論題

6. (15%) Sketch the region of integration, and evaluate the integral.∫∫R xydA where R is the region bounded by the lines y=x,y =2x,and x+ y=2.

(a)∫x2exdx(10%)

(b) ∫xex^2dx (10%)

2. Please sketch the curve (You must describe the domain, intercepts, symmetry, asymptotes, increasing or decreasing, local maximum and minimum values, and concavity and points of inflection.) (15%)

3. If f(x,y)= f(x,y) exists? Why?(10%)

4. A rectangular box without a lid is to be made from 12m 2 of cardboard. Please use the method of Lagrange multiplier to find the maximum volume of such a box.(10%)

5. Pind the radius of convergence and interval of convergence of the series .(15%)

(2) ∫e2xsin(3x) dx

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