Question Number 220963 by fantastic last updated on 21/May/25
$${Let}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:{be}\:{defined}\:{by}\:{f}\left({x},{y}\right)=\left\{\frac{{y}}{\underset{\:\:\mathrm{1},\:{y}=\mathrm{0}} {\mathrm{sin}\:{y}}},\:{y}\neq\mathrm{0}\right. \\ $$$${Then}\:{the}\:{integral}\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\underset{{x}=\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{y}=\mathrm{sin}^{−\mathrm{1}} {x}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}{f}\left({x},{y}\right){dy}\:{dx}\:{correct}\:{upto}\:{three}\:{decimal}\:{places},{is}… \\ $$
Answered by MrGaster last updated on 21/May/25
![(1/π^2 )∫_0 ^1 ∫_(sin^(−1) x) ^(π/2) (y/(sin y))dy dx =(1/π^2 )∫_0 ^(π/2) ∫_0 ^(sin y) (y/(sin y))dx dy =(1/π^2 )∫_0 ^(π/2) y dy =(1/π^2 )[(y^2 /2)]_0 ^(π/2) =(1/π^2 )∙(π^2 /8) =(1/8)≈0.125 (2):=(1/π^2 )∫_0 ^(π/2) ∫_0 ^(sin y) (y/(sin y))dx dy =(1/π^2 )∫_0 ^(π/2) y sin y∙(1/(sin y))dy =(1/π^2 )∫_(0 ) ^(π/2) y dy =(1/π^2 )[(y^2 /2)]_0 ^(π/2) =(1/π^2 )∙(π^2 /8) =(1/8)≈0.125](https://www.tinkutara.com/question/Q220967.png)
$$\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{sin}^{−\mathrm{1}} {x}} ^{\frac{\pi}{\mathrm{2}}} \frac{{y}}{\mathrm{sin}\:{y}}{dy}\:{dx} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\mathrm{sin}\:{y}} \frac{{y}}{\mathrm{sin}\:{y}}{dx}\:{dy} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {y}\:{dy} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\left[\frac{{y}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\centerdot\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}}\approx\mathrm{0}.\mathrm{125} \\ $$$$\left(\mathrm{2}\right):=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\mathrm{sin}\:{y}} \frac{{y}}{\mathrm{sin}\:{y}}{dx}\:{dy} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {y}\:\mathrm{sin}\:{y}\centerdot\frac{\mathrm{1}}{\mathrm{sin}\:{y}}{dy} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}\:} ^{\frac{\pi}{\mathrm{2}}} {y}\:{dy} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\left[\frac{{y}^{\mathrm{2}} }{\mathrm{2}}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\centerdot\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}}\approx\mathrm{0}.\mathrm{125} \\ $$