wing asked in 科學及數學數學 · 1 decade ago

Amath

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Revision Exercise 20

Q16

In Fig. 14, the shaded region is bounded by the curves y=cos x,y=sin 2x, and the x-axis within the range (x is equal or bigger that 0 and

pai/2 is smaller or equal than x)

(a) Find the coordinates of P

(b) If the region is rotated about the x-axis ,find the volume of the solid generated in term of pai.

1 Answer

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  • 1 decade ago
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    (a)

    cos x =sin 2x

    cos x =2sinxcosx

    cosx(1-2sinx)=0

    cosx=0 or sinx=1/2

    x= π/2 or π/6

    y= 0 or √3/2

    P(π/6,√3/2)

    (b)

    圖片參考:http://www.hivepc.com/calculus/survivalguide/img/d...

    The volume of the solid generated

    =π∫(cos^2 x-sin^2 2x) dx from [0 to π/6]

    =π∫(cos^2 x- 4sin^2 xcos^2 x) dx

    = π∫(4cos^4 x- 3cos^2 x) dx

    Let In=∫(cos^n x) dx

    ∫(cos^2 x) dx

    =(sin x cos x)/2+(1/2) I0 [x from 0 to π/6]

    =(1/2)(√3/2)/2+(1/2)(π/6)

    =(√3/8)+(π/12)

    ∫(cos^4 x) dx

    =(sin x cos^3 x)/4+(3/4) I2 [x from 0 to π/6]

    =(1/2)(3√3/4)(1/4)+(3/4)[(√3/8)+(π/12)]

    =(3√3/16)+(π/16)

    π∫(4cos^4 x- 3cos^2 x) dx

    =π[(3√3/4)+(π/4)-(3√3/8)-(π/4)]

    =π(3√3/8)

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