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数学：与平均有关的定积分

· 2 min read
lailai
lailai
Student & Developer

参考资料​

公式​

设：

f(t)=∫abxt+1dx∫abxtdxf(t)=\frac{\int_a^b x^{t+1}\mathrm{d}x}{\int_a^b x^t\mathrm{d}x}

当 t≠−1t\ne -1 且 t≠−2t\ne -2 时，可化简为：

f(t)=(t+1)(bt+2−at+2)(t+2)(bt+1−at+1)f(t)=\frac{(t+1)(b^{t+2}-a^{t+2})}{(t+2)(b^{t+1}-a^{t+1})}

函数 f(t)f(t) 是单调不减的。

均值​

H(a,b)≤G(a,b)≤N(a,b)≤A(a,b)≤T(a,b)H(a,b)\le G(a,b)\le N(a,b)\le A(a,b)\le T(a,b)

调和平均数（t=−3t=-3）​

f(−3)=−2(b−1−a−1)−1(b−2−a−2)=2aba+b=H(a,b)f(-3)=\frac{-2(b^{-1}-a^{-1})}{-1(b^{-2}-a^{-2})}=\frac{2ab}{a+b}=H(a,b)

几何平均数（t=−1.5t=-1.5）​

f(−1.5)=−0.5(b0.5−a0.5)0.5(b−0.5−a−0.5)=ab=G(a,b)f(-1.5)=\frac{-0.5(b^{0.5}-a^{0.5})}{0.5(b^{-0.5}-a^{-0.5})}=\sqrt{ab}=G(a,b)

海伦平均数（t=−0.5t=-0.5）​

f(−0.5)=0.5(b1.5−a1.5)1.5(b0.5−a0.5)=a+ab+b3=N(a,b)f(-0.5)=\frac{0.5(b^{1.5}-a^{1.5})}{1.5(b^{0.5}-a^{0.5})}=\frac{a+\sqrt{ab}+b}{3}=N(a,b)

算术平均数（t=0t=0）​

f(0)=1(b2−a2)2(b1−a1)=a+b2=A(a,b)f(0)=\frac{1(b^2-a^2)}{2(b^1-a^1)}=\frac{a+b}{2}=A(a,b)

质心平均数（t=1t=1）​

f(1)=2(b3−a3)3(b2−a2)=2(a2+ab+b2)3(a+b)=T(a,b)f(1)=\frac{2(b^3-a^3)}{3(b^2-a^2)}=\frac{2(a^2+ab+b^2)}{3(a+b)}=T(a,b)