mathematics-class-12-unit-07-chapter-03-definite-integral-3-7-npvlan5asro

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-1 </primary>
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- I= $ \int_{-2}^2 \frac{x^3 \sin ^4 x \cos ^2 x}{f(x)} d x $

- $ \int_{-a}^a f(x) d x=2 \int_0^a f(x) d x \text { for even function } $

- $ =0 \text { for odd function } $

- $ f(-x)=(-x)^3 \sin ^4(-x) \cos ^2(-x) $

- $ =-x^3 \sin ^4 x \cos ^2 x $

- $ =-f(x) \text { Integrand is odd. } $

- $ \text { hence } I=0 \text {. } $

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<footer style = "font-size: 18px"> $\rightarrow$ Definite integral lecture-3 $\rightarrow$ <span style = "color:blue"> Example-1 </span> $\rightarrow$ Example-2 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-2 </primary>
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- $ I=\int_0^1 \frac{\sqrt{x}}{\sqrt{x}+\sqrt{1-x}} d x \cdot-(1) \int_0^a f(x) d x=\int_0^a f(a-x) d x $

- $ I=\int_0^1 \frac{\sqrt{1-x)}}{\sqrt{1-x}+\sqrt{1-(1-x)}} d x=\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1-x}+\sqrt{x}} d x $

- (1) +(2)

- $2 I=\int_0^1 \frac{\sqrt{x}+\sqrt{1-x}}{\sqrt{1-x}+\sqrt{x}} d x $

- $ =\int_0^1 d x=1 $

- $ I=\frac{1}{2}$

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<footer style = "font-size: 18px">Definite integral lecture-3 $\rightarrow$ Example-1 $\rightarrow$ <span style = "color:blue"> Example-2 </span> $\rightarrow$ Example-3 $\rightarrow$ Example-3 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-3 </primary>
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- =2 $ \int_0^{\pi / 2} \sin ^2\left(\frac{\pi}{2}-x\right) d x $

- $I  =2 \int_0^{\pi / 2} \cos ^2 x d x-(2) $
- (1) + (2)

- $2 I=2 \int_0^{\pi / 2}\left(\sin ^2 x+\cos ^2 x\right) d x=2 x \pi$

- $2 I=\pi $

- $I =\frac{\pi}{2}$

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<footer style = "font-size: 18px">Example-2 $\rightarrow$ Example-3 $\rightarrow$ <span style = "color:blue"> Example-3 </span> $\rightarrow$ Example-4 $\rightarrow$ Example-4 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-4 </primary>
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- $ I  =\int_0^{\pi / 2} \log \left(\frac{4+3 \cos x}{4+3 \sin x}\right) d x $  (2)

- $ 2 I  =\int_0^{\pi / 2}\left[\log \left(\frac{4+3 \sin x}{4+3 \cos x}\right)+\log \left(\frac{4+3 \cos x}{4+3 \sin x}\right)\right] d x$

- $=\int_0^{\pi / 2} \log \left(\frac{4+3 \sin x}{4+\cos x} \times \frac{4 x \operatorname{sen} x}{4+\sin x}\right) d x$

- $\begin{aligned} & 2 I=\int_0^{\pi / 2} \log 1 d x \\ & \Rightarrow 2 I=0 \\\\ & I=0\end{aligned}$

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<footer style = "font-size: 18px">Example-3 $\rightarrow$ Example-4 $\rightarrow$ <span style = "color:blue"> Example-4 </span> $\rightarrow$ Example-5 $\rightarrow$ Application of definite integrals as area under simple curves </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-5 </primary>
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- $ \int_0^1 \frac{x}{I} \frac{e^x}{II} d x . $

- = $ \left.x \cdot e^x\right|_{x=0} ^{x=1}-\int_0^1 1 \cdot e^x d x $

- = $ 1 e^1-0 e^0-\left.e^x\right|_0 ^1 $

- = $ e-0-\left(e-e^0\right) $

- = $ e-e+1=1 . $

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<footer style = "font-size: 18px">Example-4 $\rightarrow$ Example-4 $\rightarrow$ <span style = "color:blue"> Example-5 </span> $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ Application of definite integrals as area under simple curves </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Application of definite integrals as area under simple curves </primary>
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- <ins>Cases.</ins>

- Area of elementary

- strip =y d x= dA

- elementary Area

- $ A=\int_{x=a}^{x=b} d A=\int_a^b y d x .$

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<footer style = "font-size: 18px">Example-4 $\rightarrow$ Example-5 $\rightarrow$ <span style = "color:blue"> Application of definite integrals as area under simple curves </span> $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ Application of definite integrals as area under simple curves </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Application of definite integrals as area under simple curves </primary>
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- elementary area

- d A=x d y

- A= $ \int_{y=c}^{y=d} d A=\int_{y=c}^{y=d} x d y . $

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<footer style = "font-size: 18px">Example-5 $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ <span style = "color:blue"> Application of definite integrals as area under simple curves </span> $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ Application of definite integrals as area under simple curves </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Application of definite integrals as area under simple curves </primary>
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- <ins>Case 3</ins>

- A= $ \int_a^b f(x) d x $

- A<0

- Required Area $=|A|$

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<footer style = "font-size: 18px">Application of definite integrals as area under simple curves $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ <span style = "color:blue"> Application of definite integrals as area under simple curves </span> $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ Example-6 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Application of definite integrals as area under simple curves </primary>
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- <ins>Case 4</ins>

- $ A_1=\int_a^c f(x) d x>0 $

- $ A_2  =\int_c^b f(x) d x<0 $

- A $ =A_1+\left|A_2\right|$

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### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-6 </primary>
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-  Area of circle

- $ x^2+y^2=a^2 $

- Total Area 4A

- $=4 \int_{x=0}^{x=a} y d x=4 \int_0^a \sqrt{a^2-x^2} d x $

- $ x^2+y^2=a^2 $

- $ y= \pm \sqrt{a^2}-\overline{x^2} $
- $ =4\left[\frac{1}{2} x \sqrt{a^2-x^2}+\frac{1}{2} a^2 \sin ^{-1} \frac{x}{a}\right]_{x=0}^{x=a} $

- $ =4\left[\frac{1}{2} a^2 \frac{\pi}{2}-0-0\right]=\pi a^2 \text {. } $
	
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<footer style = "font-size: 18px">Application of definite integrals as area under simple curves $\rightarrow$ Application of definite integrals as area under simple curves $\rightarrow$ <span style = "color:blue"> Example-6 </span> $\rightarrow$ Example-6 $\rightarrow$ Example-6 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-6 </primary>
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- Area of circle

- =4 $ \int_{y=0}^a x d y $
 
- =4 $ \int_0^a \sqrt{a^2-y^2} d y $

- $x^2+y^2=a^2$

-  x $ = \pm \sqrt{a^2-y^2} $
 
-  x $ =\sqrt{a^2-y^2} $

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<footer style = "font-size: 18px">Application of definite integrals as area under simple curves $\rightarrow$ Example-6 $\rightarrow$ <span style = "color:blue"> Example-6 </span> $\rightarrow$ Example-6 $\rightarrow$ Example-7 </footer>

<h3 id="success-stylefont-size25px-definite-integral-l-3-success"><Success style="font-size:25px"> Definite Integral L-3 </Success></h3>
<h3 id="primary-stylefont-size24px-example-6-primary"><primary style="font-size:24px"> Example-6 </primary></h3>
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<p>$ =4\left[\frac{1}{2} y \sqrt{a^2-y^2}
+\frac{1}{2} a^2 \sin ^{-1} \frac{y}{a}\right]_{y=0}^a $</p>
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<p>$ =4\left[\frac{1}{2} a^2 \frac{\pi}{2}-0-0\right]=\pi a^2 $</p>
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<footer style = "font-size: 18px">Example-6 $\rightarrow$ Example-6 $\rightarrow$ <span style = "color:blue"> Example-6 </span> $\rightarrow$ Example-7 $\rightarrow$ Example-7 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-7 </primary>
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- **Ex.** Area of Ellipse.

- Total Area

- $ =4 A \quad  =4 \int_{x=0}^{x=a} y d x $

- $ =4 \int_0^a \frac{b}{a} \sqrt{a^2-x^2} d x $

- $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \quad a>b$

- y= $\pm b \sqrt{1-\frac{x^2}{a^2}} $

- y= $ \pm \frac{b}{a} \sqrt{a^2-x^2} $

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<footer style = "font-size: 18px">Example-6 $\rightarrow$ Example-6 $\rightarrow$ <span style = "color:blue"> Example-7 </span> $\rightarrow$ Example-7 $\rightarrow$ Example-7 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-7 </primary>
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- $ =4 \int_0^b \frac{a}{b} \sqrt{b^2-y^2} d y $

- $ \frac{x^2}{a^2}=1-\frac{y^2}{b^2} $

- $ \frac{x}{a}= \pm \sqrt{1-\frac{b^2}{b^2}}$

- $ =\frac{4 a}{b}\left[\frac{1}{2} y \sqrt{b^2-y^2}+\frac{1}{2} b^2 \sin ^{-1} \frac{y}{b}\right]_0^b $

- $ =\frac{4 a}{\not b}\left[0+\frac{1}{2} b^2 \frac{\pi}{2}-0-0\right]=\pi a b,$

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<footer style = "font-size: 18px">Example-7 $\rightarrow$ Example-7 $\rightarrow$ <span style = "color:blue"> Example-7 </span> $\rightarrow$ Example-8 $\rightarrow$ Example-8 </footer>

### <Success style="font-size:25px"> Definite Integral L-3 </Success>
### <primary style="font-size:24px"> Example-8 </primary>
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- By using vertical stop (elementary rectangles)

- Required Area
 
- $=\int_{x=-1}^1 y d x-\int_{y=1}^1 y d x $

- $ y=x^2 $

- $ y=1, y=x^2 , x= \pm 1 $

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<footer style = "font-size: 18px">Example-8 $\rightarrow$ Example-8 $\rightarrow$ <span style = "color:blue"> Example-8 </span> $\rightarrow$ Example-8 $\rightarrow$ Example-9 </footer>