Question

using integration find the area of the region bounded by the triangle whose vertices are (1,0) (2,0) and (3,1)​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the given vertices as

Coordinates of A be (1, 0)

Coordinates of B be (2, 0)

Coordinates of C be (3, 1).

Now,

We know,

Equation of line passing through the points (a, b) and (c, d) is given by

$\rm :\longmapsto\:\boxed{ \tt{ \: y - b = \frac{d - b}{c - a}(x - a) \: }}$

So,

Equation of line AC passing through the points A (1, 0) and C(3, 1) is

$\rm :\longmapsto\:y - 0 = \dfrac{1 - 0}{3 - 1}(x - 1)$

$\rm :\longmapsto\:\boxed{ \tt{ \: y = \dfrac{1}{2}(x - 1) \: }} - - - (1)$

Now,

Equation of line BC passing through the points B (2, 0) and C (3, 1) is

$\rm :\longmapsto\:y - 0 = \dfrac{1 - 0}{3 - 2}(x - 2)$

$\rm :\longmapsto\:y = \dfrac{1}{1}(x - 2)$

$\rm :\longmapsto\:\boxed{ \tt{ \: y = x - 2 \: }}$

Now, from graph we concluded that,

Required area of triangle ABC is given by

$\rm \:  =  \:\displaystyle\int_1^3y_{AC} \: dx \: - \: \displaystyle\int_2^3y_{BC} \: dx$

$\rm \:  =  \:\dfrac{1}{2} \displaystyle\int_1^3 \: (x - 1)dx \: - \: \displaystyle\int_2^3(x - 2) \: dx$

We know,

$\rm :\longmapsto\:\boxed{ \tt{ \: \displaystyle\int \: {x}^{n} \: dx \: = \: \frac{ {x}^{n + 1} }{n + 1} + c \: }}$

So, using this,

$\rm \:  =  \:\dfrac{1}{2}\bigg[\dfrac{ {x}^{2} }{2} - x\bigg]_1^3 - \bigg[\dfrac{ {x}^{2} }{2} - 2x\bigg]_2^ 3$

$\rm \:=\dfrac{1}{2}\bigg[\dfrac{9 - 1}{2} - (2 - 1)\bigg] - \bigg[\dfrac{ (9 - 4)}{2} - 2(3 - 2)\bigg]$

$\rm \:=\dfrac{1}{2}\bigg[\dfrac{8}{2} - (1)\bigg] - \bigg[\dfrac{5}{2} - 2(1)\bigg]$

$\rm \:=\dfrac{1}{2}\bigg[4 - 1\bigg] - \bigg[\dfrac{5}{2} - 2\bigg]$

$\rm \:=\dfrac{1}{2}\bigg[3\bigg] - \bigg[\dfrac{5 - 4}{2}\bigg]$

$\rm \:=\dfrac{3}{2} - \bigg[\dfrac{1}{2}\bigg]$

$\rm \:  =  \:1 \: square \: units$

More to know :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$