- #1
funcalys
- 30
- 1
We have that
[itex]\int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}\int^{1}_{t}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}[arcsin(x)]^{1}_{t}=\frac{\pi}{2}[/itex]
However, I think [itex]\int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}[/itex] should equal to [itex]lim_{\stackrel{}{t \rightarrow 1^{-}}}\int^{t}_{0}\frac{1}{\sqrt{1-x^{2}}}[/itex]
since f is not continuous at 1, not 0.
[itex]\int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}\int^{1}_{t}\frac{1}{\sqrt{1-x^{2}}}=lim_{\stackrel{}{t \rightarrow 0^{+}}}[arcsin(x)]^{1}_{t}=\frac{\pi}{2}[/itex]
However, I think [itex]\int^{1}_{0}\frac{1}{\sqrt{1-x^{2}}}[/itex] should equal to [itex]lim_{\stackrel{}{t \rightarrow 1^{-}}}\int^{t}_{0}\frac{1}{\sqrt{1-x^{2}}}[/itex]
since f is not continuous at 1, not 0.