- #1
ManishR
- 88
- 0
[tex]\intop_{a}^{b}f(x)dx=\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}f\left(a+i\frac{(b-a)}{h}\right)\frac{(b-a)}{h}[/tex]
if f(x) = x
[tex]\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(a+i\frac{(b-a)}{h})\frac{1}{h}\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{a}{h}+i\frac{(b-a)}{h^{2}})\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{a}{h}+i\frac{(b-a)}{h^{2}})\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+\left(\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(i\frac{(b-a)}{h^{2}})\right)\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+(b-a)\left(\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{i}{h^{2}})\right)\right][/tex]
let
[tex]\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{i}{h^{2}})=k[/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+(b-a)k\right][/tex]
if
[tex]\intop_{a}^{b}f(x)dx=b^{2}-a^{2}[/tex]
[tex]\Rightarrow b^{2}-a^{2}=(b-a)\left[a+(b-a)k\right][/tex]
[tex]\Rightarrow k=\frac{b}{b-a}[/tex]
but k cannot be function of anything.
so what's wrong here ?
if f(x) = x
[tex]\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(a+i\frac{(b-a)}{h})\frac{1}{h}\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{a}{h}+i\frac{(b-a)}{h^{2}})\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{a}{h}+i\frac{(b-a)}{h^{2}})\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+\left(\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(i\frac{(b-a)}{h^{2}})\right)\right][/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+(b-a)\left(\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{i}{h^{2}})\right)\right][/tex]
let
[tex]\underset{h\rightarrow\infty}{lim}\sum_{0}^{h}(\frac{i}{h^{2}})=k[/tex]
[tex]\Rightarrow\intop_{a}^{b}f(x)dx=(b-a)\left[a+(b-a)k\right][/tex]
if
[tex]\intop_{a}^{b}f(x)dx=b^{2}-a^{2}[/tex]
[tex]\Rightarrow b^{2}-a^{2}=(b-a)\left[a+(b-a)k\right][/tex]
[tex]\Rightarrow k=\frac{b}{b-a}[/tex]
but k cannot be function of anything.
so what's wrong here ?