Name: Area Between Two Curves - Horizontal Strip Method
Uploaded: 2019-07-19
Duration: 3 min 41 s
Description: We've learnt how to find the area bound by two curves using the vertical strip method. Can we also find the area bound by two curves using the horizontal str...

Question

Find the area of the region included between the parabola

y^{2} = x

and the line

x + y = 2

.

Accepted Answer

Given the equation of curves are

$y^{2} = x . . . (1)$

$x + y = 2 . . . (2)$

Solving $(1)$ and $(2)$ , we have

$y^{2} = 2 - y \Rightarrow y^{2} + y - 2 = 0$

$\Rightarrow (y + 2) (y - 1) = 0$

$\Rightarrow y = - 2, 1$

$\Rightarrow x = 4, 1$

Equation $(1)$ represents a parabola with vertex at the origin and its axis as $x$ -axis, equation $(2)$ represents a line passing through $(2, 0)$ and $(0, 2)$ . points of intersection of line and parabola are $(1, 1)$ and $(4, - 2)$ .

A rough sketch of curves is as below:-

The shaded region represents the required area. We slice it in rectangles of width $△ y$ and length $= (x_{1} - x_{2})$ .

Area of rectangle $= (x_{1} - x_{2}) △ y$

This approximation rectangle slides from $y = - 2$ to $y = 1$ , so

Required area $=$ Region $A O B A$

$= \int_{- 2}^{1} (x_{1} - x_{2}) d y$

$= \int_{- 2}^{1} (2 - y - y^{2}) d y$

$= {[2 y - \frac{y^{2}}{2} - \frac{y^{3}}{3}]}_{- 2}^{1}$

$= [(2 - \frac{1}{2} - \frac{1}{3}) - (- 4 - 2 + \frac{8}{3})]$

$= \frac{7}{6} + \frac{10}{3}$

Required area $= \frac{9}{2} sq . units$ .

Find the area of the region included between the parabola $y^{2} = x$ and the line $x + y = 2$ .

Important Questions on Areas of Bounded Regions

Learn and Prepare for any exam you want !

Find the area of the region included between the parabola y2=x and the line x+y=2.

Important Questions on Areas of Bounded Regions

Learn and Prepare for any exam you want !

Find the area of the region included between the parabola $y^{2} = x$ and the line $x + y = 2$ .