Mathematics

# Evaluate $\displaystyle\int_{0}^{a}\sqrt{a^{2}-x^{2}}dx$

##### SOLUTION
$I=\displaystyle\int_{0}^{a}\sqrt{a^{2}-x^{2}}dx$
Let $x=a\sin t$
Then, $dx=a\cos t dt$                               [ Differentiating both sides ]
When $x=0,$
$0=a\sin t$
$t=\sin^{-1}(0)$
$\therefore$ $t=0$
When, $x=a,$
$a=a\sin t$
$1=\sin t$
$t=\sin^{-1}(1)$
$\therefore$  $t=\dfrac{\pi}{2}$

$I=\displaystyle\int_{0}^{a}\sqrt{a^{2}-x^{2}}dx$

$\Rightarrow$  $I=\displaystyle\int_0^{\dfrac{\pi}{2}}\sqrt{(a^2-a^2\sin^2t)}a\cos t dt$

$=\displaystyle\int_0^{\dfrac{\pi}{2}}\sqrt{a^2(1-\sin^2t)}a\cos t dt$

$=\displaystyle\int_0^{\dfrac{\pi}{2}}\sqrt{a^2\cos^2 t}a\cos t dt$

$=\displaystyle\int_0^{\dfrac{\pi}{2}}a\cos t.a\cos t dt$

$=\displaystyle\int_0^{\dfrac{\pi}{2}}a^2\cos^2 t dt$

$=a^2\displaystyle\int_0^{\dfrac{\pi}{2}} \dfrac{1+\cos 2t}{2}dt$

$=\dfrac{a^2}{2}\left[t+\dfrac{\sin 2t}{2}\right]^{\dfrac{\pi}{2}}_0$

$=\dfrac{a^2}{2}\left(\dfrac{\pi}{2}-0\right)$

$=\dfrac{a^2\pi}{4}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int^{\pi}_0 \frac{xdx}{a^2\cos^2x+b^2\,\sin^2x}$
• A. $\dfrac{\pi}{2ab}$
• B. $\dfrac{\pi}{ab}$
• C. $\dfrac{\pi^2}{2ab}$
• D. $\dfrac{\pi^2}{ab}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Let $A=\displaystyle\int _{ 1 }^{ { e }^{ 2 } }{ \dfrac { \ell nx }{ \sqrt { x } } dx }$, then ?
• A. $A > 2\left( e-\dfrac { 1 }{ e } \right)$
• B. $A > \left( e-1 \right) \left( 2+\dfrac { 1 }{ \sqrt { e } } \right)$
• C. $A > \left( { e }^{ 2 }-1 \right) \dfrac { 2 }{ e }$
• D. $A < \left( e-1 \right) \left( 2+\dfrac { 1 }{ \sqrt { e } } \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate: $\displaystyle \int \frac{5x^{8}+7x^{6}}{\left ( x^{2}+1+2x^{7} \right )^{2}}dx$
• A. $\displaystyle \frac{2x^{7}}{2x^{7}+x^{2}+1}$
• B. $\displaystyle \frac{x^{6}}{2x^{7}+x^{2}+1}$
• C. $\displaystyle \frac{x^{14}}{2x^{7}+x^{2}+1}$
• D. $\displaystyle \frac{x^{7}}{2x^{7}+x^{2}+1}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral
$\int { \cfrac { 1 }{ x\log { x } } } dx$

$\int \frac{1}{1+x}\;dx$