設

棱臺

等高截面等積圓臺

母線與對稱軸所成角為θ

有

r/h=tanθ

即

r2/h2=tan2θ

即

πr2/h2=πtan2θ

即

S/h2=πtan2θ

(定值πtan2θ即這位網(wǎng)友所設常數(shù)k)

即

h=√S/(√πtanθ)

即

dh=1/(2√(πS)tanθ)dS

即

V

=

∫(h1，h2)Sdh

=

∫(S1，S2)S/(2√(πS)tanθ)dS

=

∫(S1，S2)√S/(2√πtanθ)dS

=

1/(2√πtanθ)∫(S1，S2)√SdS

=

1/(2√πtanθ)·2/3S^(3/2)|(S1，S2)

=

1/(3√πtanθ)(√S23-√S13)

=

(√S2-√S1)

(S1+S2+√(S1S2))

/

(3√πtanθ)

=

(h2-h1)

(S1+S2+√(S1S2))

/

3

=

H(S1+S2+√(S1S2))/3

ps.

抑或

(1)

有

S1/S2

=

h12/h22

即

S1/h12=S2/h22

設

S1/h12=S2/h22=k

有

S1=kh12

S2=kh22

即

S=kh

(2)

V

=

∫(h1，h2)Sdh

=

∫(h1，h2)πtan2θh2dh

=

πtan2θ∫(h1，h2)h2dh

=

πtan2θ/3·h3|(h1，h2)

=

πtan2θ(h23-h13)/3

=

πtan2θ(h2-h1)(h12+h1h2+h22)/3

=

πH(r12+r1r2+r22)/3

=

H(S1+√(S1S2)+S2)/3