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100 台灣大學微積分(C) 第 11 題

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100學年度 · 100台大微積分C · 第 11 題

題目

11. Let AA be the region bounded above by the graph of y=sin(πx)y=\sin(\pi x) and below by the graph of y=x34xy=x^3-4x.

(A) (2.1,0.8)(2.1,0.8) falls into the region AA.

(B) The area of AA can be represented by 02[sin(πx)(x34x)]dx\displaystyle \int_0^2[\sin(\pi x)-(x^3-4x)]\,dx.

(C) The area of AA is equal to 4.

(D) The region AA is the base of a solid. For this solid, each cross section perpendicular to the xx-axis is a square. The volume of this solid is equal to 02[sin(πx)(x34x)]2dx\displaystyle \int_0^2[\sin(\pi x)-(x^3-4x)]^2\,dx.

(E) The region AA models the surface of a small pond. At all points in AA at a distance xx from the yy-axis, the depth of the water is given by h(x)=3xh(x)=3-x. The volume of water in the pond is equal to 02(3x)[sin(πx)(x34x)]dx\displaystyle \int_0^2(3-x)[\sin(\pi x)-(x^3-4x)]\,dx.

解答