Integration Questions

Question Number 154142 by EDWIN88 last updated on 14/Sep/21

Answered by mr W last updated on 14/Sep/21

$${y}={a}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right)^{\mathrm{2}} \\$$$$\mathrm{8}={a}\left(−\mathrm{4}\right)\left(+\mathrm{2}\right)^{\mathrm{2}} \:\Rightarrow{a}=−\frac{\mathrm{1}}{\mathrm{2}} \\$$$${y}=−\frac{\mathrm{1}}{\mathrm{2}}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right)^{\mathrm{2}} \\$$$$\frac{{dy}}{{dx}}=−\frac{\mathrm{1}}{\mathrm{2}}\left[\left({x}+\mathrm{2}\right)^{\mathrm{2}} +\mathrm{2}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right)\right]=\mathrm{0} \\$$$$\left({x}+\mathrm{2}\right)\left({x}−\mathrm{2}\right)=\mathrm{0} \\$$$$\Rightarrow{x}=−\mathrm{2}\:\left({rejected}\right) \\$$$$\Rightarrow{x}=\mathrm{2}\:\Rightarrow{x}_{{P}} =\mathrm{2},\:{y}_{{P}} =−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}−\mathrm{4}\right)\left(\mathrm{2}+\mathrm{2}\right)^{\mathrm{2}} =\mathrm{16} \\$$$${eqn}.\:{of}\:{OP}: \\$$$${y}=\mathrm{8}{x} \\$$$${eqn}.\:{of}\:{QR}: \\$$$${y}=\mathrm{8}−\frac{{x}}{\mathrm{8}} \\$$$$\mathrm{8}−\frac{{x}}{\mathrm{8}}=\mathrm{8}{x}\:\Rightarrow{x}_{{R}} =\frac{\mathrm{64}}{\mathrm{65}} \\$$$${A}=\int_{\mathrm{0}} ^{\mathrm{4}} {ydx}−\frac{\mathrm{1}}{\mathrm{2}}×\mathrm{8}×\frac{\mathrm{64}}{\mathrm{65}} \\$$$${A}=−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{4}} \left({x}−\mathrm{4}\right)\left({x}+\mathrm{2}\right)^{\mathrm{2}} {dx}−\frac{\mathrm{256}}{\mathrm{65}} \\$$$${A}=−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{4}} \left({x}^{\mathrm{3}} −\mathrm{12}{x}−\mathrm{16}\right){dx}−\frac{\mathrm{256}}{\mathrm{65}} \\$$$${A}=−\frac{\mathrm{1}}{\mathrm{2}}\left[\frac{\mathrm{4}^{\mathrm{4}} }{\mathrm{4}}−\mathrm{6}×\mathrm{4}^{\mathrm{2}} −\mathrm{16}×\mathrm{4}\right]−\frac{\mathrm{256}}{\mathrm{65}} \\$$$${A}=\mathrm{48}−\frac{\mathrm{256}}{\mathrm{65}}=\frac{\mathrm{2864}}{\mathrm{65}}\approx\mathrm{44}.\mathrm{06} \\$$

Commented by iloveisrael last updated on 15/Sep/21

$${yes} \\$$

Answered by iloveisrael last updated on 15/Sep/21