有没有人想过,圆柱体体积公式是怎么来的?
读过小学的都知道教科书上用的是倒水法,但是太不标准了!!!
废话不多说直接开始:
s=设DA=r,CB=R,EB=H,AE=h,绿色圆的面积=S
rh=RHr=RhHS=π(RhH)2∫0HS dh=∫0Hπ(RhH)2 dh=π∫0H(RhH)2 dh=πR2H2∫0Hh2 dh\frac{r}{h}=\frac{R}{H}\\ r=\frac{Rh}{H}\\ S=\pi(\frac{Rh}{H})^2\\ \int_{0}^{H}{S}\ dh=\int_0^{H}\pi(\frac{Rh}{H})^2\ dh\\ =\pi\int_0^H{(\frac{Rh}{H})^2}\ dh\\ =\pi\frac{R^2}{H^2}\int_{0}^{H}{h}^2\ dh\\ hr =HR r=HRh S=π(HRh )2∫0H
S dh=∫0H π(HRh )2 dh=π∫0H (HRh )2 dh=πH2R2 ∫0H h2 dh
=πR2H2[h33]0H=πR2H2⋅(H33−033)=πR2H3=\pi\frac{R^2}{H^2}[\frac{h^3}{3}]_0^H\\ =\pi\frac{R^2}{H^2}\cdot (\frac{H^3}{3}-\frac{0^3}{3})\\ =\pi R^2 \frac{H}{3}\\ =πH2R2 [3h3 ]0H =πH2R2 ⋅(3H3 −303 )=πR23H
∫anxndx=an+1n+1−bn+1n+1\int_a^nx^ndx=\frac{a^{n+1}}{n+1}-\frac{b^{n+1}}{n+1} ∫an xndx=n+1an+1 −n+1bn+1
证明完成!!!!!
一键三连求求了(虽然不是b站)
是不是很简单!!!
有心就加入我们吧!
