Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 31141 by Cheyboy last updated on 03/Mar/18

using the limit defination find the area of f(x)= cos(x) [0,π/2]

$$\boldsymbol{{using}}\:\boldsymbol{{the}}\:\boldsymbol{{limit}}\:\boldsymbol{{defination}} \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{area}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\:\:\left[\mathrm{0},\pi/\mathrm{2}\right] \\ $$

Answered by Joel578 last updated on 03/Mar/18

A = ∫_0 ^(π/2) cos x dx = lim_(n→∞) Σ_(i=1) ^n f(x_i )Δx_i = lim_(n→∞) Σ_(i=1) ^n cos (((iπ)/(2n))) (π/(2n)) = lim_(n→∞) (π/(2n)) Σ_(i=1) ^n cos (((iπ)/(2n))) = (π/2) lim_(n→∞) ((Σ_(i=1) ^n cos (((iπ)/(2n))))/n) = (π/2) lim_(n→∞) ((cos ((π/(2n))) + cos ((π/n)) + cos (((3π)/(2n))) + ... + cos (((nπ)/(2n))))/n)

$${A}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}\:{x}\:{dx}\:=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:{f}\left({x}_{{i}} \right)\Delta{x}_{{i}} \\ $$$$\:\:\:\:\:=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\left(\frac{{i}\pi}{\mathrm{2}{n}}\right)\:\frac{\pi}{\mathrm{2}{n}} \\ $$$$\:\:\:\:\:=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi}{\mathrm{2}{n}}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\left(\frac{{i}\pi}{\mathrm{2}{n}}\right) \\ $$$$\:\:\:\:\:=\:\frac{\pi}{\mathrm{2}}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\left(\frac{{i}\pi}{\mathrm{2}{n}}\right)}{{n}} \\ $$$$\:\:\:\:\:=\:\frac{\pi}{\mathrm{2}}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}{n}}\right)\:+\:\mathrm{cos}\:\left(\frac{\pi}{{n}}\right)\:+\:\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}{n}}\right)\:+\:...\:+\:\mathrm{cos}\:\left(\frac{{n}\pi}{\mathrm{2}{n}}\right)}{{n}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com