佑信 發問時間: 科學數學 · 9 年前

Calculus - areas in polar ....

Find the area of the region that lies inside the

circle r= 1/2 and inside the four-leaved rose r= cos2θ.

圖片參考:http://imgcld.yimg.com/8/n/AB08436180/o/1612042704...

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  • 麻辣
    Lv 7
    9 年前
    最佳解答

    Find the area of the region that lies inside the

    circle r= 1/2 and inside the four-leaved rose r= cos2θ.dA=0.5*r^2*dθ=0.5*(cosθ-1/2)^2*dθA=4*∫(cos2θ-1/2)^2*dθ.....θ=0 to 30°=4∫(cos2θ^2-cos2θ+1/4)*dθ=4∫[(1+cos4θ)/2-cos2θ+1/4]*dθ=4∫[cos4θ/2-cos2θ+3/4]dθ=2∫cos4θdθ-4∫cos2θ*dθ+3θ=sin4θ/2-2sin2θ+3θ=0.5*sin120°-2sin60°+3*π/6-0-0-0=√3/4-√3+π/2=π/2-√3/4............ans

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