109年 - 109 元智大學轉學生招生考試試題_機械工程學系：微積分#114194

(a) (10%)

(b)∫ x arcsec x dx (10%)

2. (15%) Find the centroid of the region bounded by the graph of the parametric equations given by x =  = 4 - t and the coordinate axes.

3. (10 %) Find the area of the surface formed by revolving the polar equation r = 2 cos θ over the interval about the line r = 5 secθ.

5. (10%) Find the area of the portion of the surface given by f(x,y) = 1 - x² +y that lies above the triangular region R with vertices (1, 0, 0), (0, -1, 0), and (0, 1, 0).

6. (10 %) Use a change of variables to evaluate the double integral dA.

7. (10 %) Find the extreme values of the function f(x,y)= x² + 2y² -2x +3 subject to the constraint x² +y² ≤10.

8. (15 %) Find a power series for f(x) = arctan x centered at 0, and use it to approximate the value dx, using R_N ≤0.001.