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 133541 by bemath last updated on 22/Feb/21

∫_0 ^3 (√(9−x^2 )) dx =?  (a)13.5      (b)21      (c)22.5  (d)1.8         (e) 30

$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\sqrt{\mathrm{9}−\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$$$\left(\mathrm{a}\right)\mathrm{13}.\mathrm{5}\:\:\:\:\:\:\left(\mathrm{b}\right)\mathrm{21}\:\:\:\:\:\:\left(\mathrm{c}\right)\mathrm{22}.\mathrm{5} \\ $$$$\left(\mathrm{d}\right)\mathrm{1}.\mathrm{8}\:\:\:\:\:\:\:\:\:\left(\mathrm{e}\right)\:\mathrm{30} \\ $$

Commented by MJS_new last updated on 23/Feb/21

it′s a quarter circle of radius 3 ⇒ answer is ((9π)/4)≈7.06858

$$\mathrm{it}'\mathrm{s}\:\mathrm{a}\:\mathrm{quarter}\:\mathrm{circle}\:\mathrm{of}\:\mathrm{radius}\:\mathrm{3}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{9}\pi}{\mathrm{4}}\approx\mathrm{7}.\mathrm{06858} \\ $$

Commented by bemath last updated on 23/Feb/21

it meant all available answer  is wrong.

$$\mathrm{it}\:\mathrm{meant}\:\mathrm{all}\:\mathrm{available}\:\mathrm{answer} \\ $$$$\mathrm{is}\:\mathrm{wrong}.\: \\ $$

Answered by mathmax by abdo last updated on 23/Feb/21

∫_0 ^3 (√(9−x^2 ))dx =_(x=3sint)   3∫_0 ^(π/2) cost(3cost)dt =9∫_0 ^(π/2)  cos^2 t dt  =(9/2)∫_0 ^(π/2) (1+cos(2t))dt =((9π)/4)+(9/4)[sin(2t)]_0 ^(π/2)  =((9π)/4)

$$\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{9}−\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:=_{\mathrm{x}=\mathrm{3sint}} \:\:\mathrm{3}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{cost}\left(\mathrm{3cost}\right)\mathrm{dt}\:=\mathrm{9}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{2}} \mathrm{t}\:\mathrm{dt} \\ $$$$=\frac{\mathrm{9}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}+\mathrm{cos}\left(\mathrm{2t}\right)\right)\mathrm{dt}\:=\frac{\mathrm{9}\pi}{\mathrm{4}}+\frac{\mathrm{9}}{\mathrm{4}}\left[\mathrm{sin}\left(\mathrm{2t}\right)\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:=\frac{\mathrm{9}\pi}{\mathrm{4}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com