Maths 1 Marks

2. $\pi\text{ sq.}\text{ units}$

y² = 8x represents a parabola with vertex at origin and axis of symmetry a long the +ve direction of x-axis 

x = 2 is line parallel to y-axis.

Let (x, y) be a given point o the parabola, y² = 8x

Since parabola y² = 8x is symmetric about x-axis,

$\therefore$ Required area = 2(area OCAO)

On slicing the area above x-axis into vertical strios of length = |y| and width = dx

⇒ area of rectangular strip = |y|dx

The approximating rectangle moves between x = 0 and x = 2

So, area A = 2

$\Rightarrow \text{A}=2\int\limits_{0}^{2}\mid\text{y}\mid\text{dx}=2\int\limits_{0}^{2}\text{y}\text{dx}$

$\Rightarrow \text{A}=2\int\limits_{0}^{2}\sqrt{8\text{x}}\text{dx}$

$\Rightarrow \text{A}=2\times2\int\limits_{0}^{2}\sqrt{2\text{x}}\text{dx}=4\sqrt{2}\int\limits_{0}^{2}\sqrt{\text{x}}\text{dx}$

$\Rightarrow \text{A}=4\sqrt{2}\Bigg[\frac{\text{x}^\frac{3}{2}}{\frac{3}{2}}\Bigg]^{2}_{0}=\frac{8}{3}\sqrt{2}\Big[2^\frac{3}{2}-0\Big]$

$=\frac{8}{3}\times2^{2}=\frac{32}{3}$

$\therefore$ Area $\text{A}=\frac{32}{3}\ \text{sq}.\ \text{units}$

$\text{y}=\sqrt{\text{x}+1}$ in [0, 4] representa a curve which is part of a parabola x = 4 represrnts a line parallel to y-axis and cutting x-axis at (0, 4) 

Enclosed area bound by the cure and lines x = 0 and x = 4 is OABCO consider a vertical strip of length = |y| and width = dx

$\therefore$ Area of appoximating rectangle moves from x = 0 to x = 4

$\Rightarrow \text{A}=\text{Area OABCO}=\int\limits_{0}^{4} |\text{y}|\text{dx}$

$\Rightarrow \text{A}=\int\limits_{0}^{4} \text{y}\text{dx}$

$\Rightarrow \text{A}=\int\limits_{0}^{4}\sqrt{\text{x}+1}\text{dx}$

$\Rightarrow \text{A}=\int\limits_{0}^{4}({\text{x}+1})^\frac{1}{2}\text{dx}$

$\Rightarrow \text{A}=\Bigg[\frac{(\text{x}+1)^\frac{3}{2}}{\frac{3}{2}}\Bigg]^{4}_{0}$

$\Rightarrow \text{A}=\frac{2}{3}\Big(5^\frac{3}{2}-1\Big)\ \text{sq.}\ \text{units}$

$\therefore$ Enclosed area between the cure and given lines $\text{A}=\frac{2}{3}\Big(5^\frac{3}{2}-1\Big)\ \text{sq.}\ \text{units}$