Notes: `y = tan(x), y = 2sin(x), (-pi/3)

Tuesday, 22 November 2016

$\displaystyle{y}={\tan{{\left({x}\right)}}},{y}={2}{\sin{{\left({x}\right)}}},{\left(-\frac{\pi}{{3}}\right)}$

$\displaystyle{y}={\tan{{\left({x}\right)}}},{y}={2}{\sin{{\left({x}\right)}}},-\frac{\pi}{{3}}\le{x}\le\frac{\pi}{{3}}$

Refer the attached image. Graph of y=tan(x) is plotted in blue color and graph of y=2sin(x) is plotted in red color.

Area of the region enclosed by the given curves(A)= $\displaystyle{2}{\int_{{0}}^{{\frac{\pi}{{3}}}}}{\left({2}{\sin{{\left({x}\right)}}}-{\tan{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$ (by symmetry)

$\displaystyle{A}={2}{{\left[-{2}{\cos{{\left({x}\right)}}}-{\ln{{\sec{{\left({x}\right)}}}}}\right]}_{{0}}^{{\frac{\pi}{{3}}}}}$

$\displaystyle{A}={2}{\left({\left(-{2}{\cos{{\left(\frac{\pi}{{3}}\right)}}}-{\ln{{\sec{{\left(\frac{\pi}{{3}}\right)}}}}}\right)}-{\left(-{2}{\cos{{\left({0}\right)}}}-{\ln{{\sec{{\left({0}\right)}}}}}\right)}\right)}$

$\displaystyle{A}={2}{\left({\left(-{2}\cdot{\left(\frac{{1}}{{2}}\right)}-{\ln{{2}}}\right)}-{\left(-{2}-{0}\right)}\right)}$

$\displaystyle{A}={2}{\left({1}-{\ln{{2}}}\right)}$

$\displaystyle{y}={\tan{{\left({x}\right)}}},{y}={2}{\sin{{\left({x}\right)}}},-\frac{\pi}{{3}}\le{x}\le\frac{\pi}{{3}}$

Refer the attached image. Graph of y=tan(x) is plotted in blue color and graph of y=2sin(x) is plotted in red color.

$\displaystyle{A}={2}{{\left[-{2}{\cos{{\left({x}\right)}}}-{\ln{{\sec{{\left({x}\right)}}}}}\right]}_{{0}}^{{\frac{\pi}{{3}}}}}$

$\displaystyle{A}={2}{\left({\left(-{2}\cdot{\left(\frac{{1}}{{2}}\right)}-{\ln{{2}}}\right)}-{\left(-{2}-{0}\right)}\right)}$

$\displaystyle{A}={2}{\left({1}-{\ln{{2}}}\right)}$

Notes

Tuesday, 22 November 2016

$\displaystyle{y}={\tan{{\left({x}\right)}}},{y}={2}{\sin{{\left({x}\right)}}},{\left(-\frac{\pi}{{3}}\right)}$

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Is there any personification in "The Tell-Tale Heart"?

Tuesday, 22 November 2016

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Is there any personification in &quot;The Tell-Tale Heart&quot;?

Is there any personification in "The Tell-Tale Heart"?