Notes: `y = cos(x), y = 1

Thursday, 31 October 2013

$\displaystyle{y}={\cos{{\left({x}\right)}}},{y}={1}-{\cos{{\left({x}\right)}}},{0}$

$\displaystyle{y}={\cos{{\left({x}\right)}}},{y}={1}-{\cos{{\left({x}\right)}}},{0}\le{x}\le\pi$

Refer the attached image, y=cos(x) is plotted in red color and y=1-cos(x) is plotted in blue color.

From the graph,

cos(x) is above (1-cos(x)) from 0 to pi/3 and

(1-cos(x)) is above cos(x) from pi/3 to pi.

Area of the region enclosed by the given curves A= $\displaystyle{\int_{{0}}^{{\frac{\pi}{{3}}}}}{\left({\cos{{\left({x}\right)}}}-{\left({1}-{\cos{{\left({x}\right)}}}\right)}\right)}{\left.{d}{x}\right.}+{\int_{{\frac{\pi}{{3}}}}^{\pi}}{\left({\left({1}-{\cos{{\left({x}\right)}}}\right)}-{\cos{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$

$\displaystyle{A}={\int_{{0}}^{{\frac{\pi}{{3}}}}}{\left({2}{\cos{{\left({x}\right)}}}-{1}\right)}{\left.{d}{x}\right.}+{\int_{{\frac{\pi}{{3}}}}^{\pi}}{\left({1}-{2}{\cos{{\left({x}\right)}}}\right)}{\left.{d}{x}\right.}$

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

$\displaystyle{A}={\left({\left({2}{\sin{{\left(\frac{\pi}{{3}}\right)}}}-\frac{\pi}{{3}}\right)}-{\left({2}{\sin{{\left({0}\right)}}}-{0}\right)}\right)}+{\left(\pi-{2}{\sin{{\left(\pi\right)}}}-{\left(\frac{\pi}{{3}}-{2}{\sin{{\left(\frac{\pi}{{3}}\right)}}}\right)}\right.}$

$\displaystyle{A}={\left({2}\cdot\frac{\sqrt{{{3}}}}{{2}}-\frac{\pi}{{3}}\right)}+\pi-\frac{\pi}{{3}}+{2}\cdot\frac{\sqrt{{{3}}}}{{2}}$

$\displaystyle{A}=\sqrt{{{3}}}-\frac{\pi}{{3}}+\pi-\frac{\pi}{{3}}+\sqrt{{{3}}}$

$\displaystyle{A}=\frac{\pi}{{3}}+{2}\sqrt{{{3}}}\approx{4.511}$

$\displaystyle{y}={\cos{{\left({x}\right)}}},{y}={1}-{\cos{{\left({x}\right)}}},{0}\le{x}\le\pi$

Refer the attached image, y=cos(x) is plotted in red color and y=1-cos(x) is plotted in blue color.

From the graph,

cos(x) is above (1-cos(x)) from 0 to pi/3 and

(1-cos(x)) is above cos(x) from pi/3 to pi.

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

$\displaystyle{A}={\left({2}\cdot\frac{\sqrt{{{3}}}}{{2}}-\frac{\pi}{{3}}\right)}+\pi-\frac{\pi}{{3}}+{2}\cdot\frac{\sqrt{{{3}}}}{{2}}$

$\displaystyle{A}=\sqrt{{{3}}}-\frac{\pi}{{3}}+\pi-\frac{\pi}{{3}}+\sqrt{{{3}}}$

$\displaystyle{A}=\frac{\pi}{{3}}+{2}\sqrt{{{3}}}\approx{4.511}$

Notes

Thursday, 31 October 2013

$\displaystyle{y}={\cos{{\left({x}\right)}}},{y}={1}-{\cos{{\left({x}\right)}}},{0}$

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

Thursday, 31 October 2013

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

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