Question
Using integration, Find the area bounded by the the triangle whose vartices are $(2, 1), (3, 4)$ and $(5, 2)$.
✓
Answer
Consider the points A(2, 1), B(3, 4) and C(5, 2)
We need to find area of shaded triangle ABC
Equation of AB is
$\text{y}-1=\Big(\frac{4-3}{3-2}\Big)(\text{x}-2)$
$\Rightarrow\text{x}-3\text{y}-5=0\ ...(\text{i})$
Equation of BC is
$\text{y}-4=\Big(\frac{2-4}{5-3}\Big)(\text{x}-3)$
$\Rightarrow\text{x}=\text{y}-7=0\ ...(\text{ii})$
Equation of CA is
$\text{y}-2=\Big(\frac{2-1}{5-2}\Big)(\text{x}-5)$
$\Rightarrow\text{x}-3\text{y}+2=0\ ...(\text{iii})$
Area of $\triangle\text{ABC}= \text{Area of}\ \triangle\text{ABC}+ \text{Area of }\triangle\text{ABC}$ in the,
Consided point $P(x_1, y_2)$ on AB and $Q(x_1, y_1)$ on AD
Thus, the area of appoximating with length = $|y_2 - y_1|$ from x = 2, to x = 3
$\therefore$ Area of $$$\triangle\text{ABC}=\int\limits_{2}^{3}|\text{y}_{2}-\text{y}_{1}|\text{dx} $
$\Rightarrow\text{A}=\int\limits_{2}^{3}(\text{y}_{2}-\text{y}_{1})\text{dx} $
$\Rightarrow\text{A}=\int\limits_{2}^{3}(3\text{x}-5)-\Big(\frac{\text{x}-1}{3}\Big)\text{dx}$
$\Rightarrow\text{A}=\int\limits_{2}^{3}\frac{(9\text{x}-15-\text{x}-1)}{3}\text{dx}$
$\Rightarrow\text{A}=\int\limits_{2}^{3}\frac{(8\text{x}-16)}{3}\text{dx}$
$\Rightarrow\text{A}=\frac{1}{3}\Big[8\text{x}^{2}-16\text{x}\Big]\text{dx}$
$\Rightarrow\text{A}=\frac{1}{3}(68-64)$
$\Rightarrow\text{A}=\frac{4}{3}\ \text{sq.}\ \text{units}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Fine: $\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{dx}.$
→At what points on the interval $[0,\ 2\pi]$ does the function sin 2x attain its maximum value?
→Find the maximum and minimum values of the function $\text{f}(\text{x})=\frac{4}{\text{x}+2}+\text{x}$
→Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio $\text{1 : 2 : 4. }$The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.
→Differentiate $\tan^{-1}\bigg(\frac{\sqrt{1 - \text{x}^{2}}}{\text{x}}\bigg)$with respect to $\cos^{-1}(2 \text{x}\sqrt{1- \text{x}^{2}}),$ when $\text{x}\neq0.$
→Find the equation of the perpendicular drawn from the point P(-1, 3, 2) to the line $\vec{\text{r}}=\big(2\hat{\text{j}}+3\hat{\text{k}}\big)+\lambda\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the coordinates of the foot of the perpendicular from P.
→Solve the following systems of linear equations by cramer's rule:
→$x + y = 5,$
$y + z = 3,$
$x + z = 4$
Evaluate the following integrals:
$\int\frac{\text{x}^2+1}{\text{x}^2+7\text{x}^2+1}\ \text{dx}$
→$\int\frac{\text{x}^2+1}{\text{x}^2+7\text{x}^2+1}\ \text{dx}$
Show that the following systems of linear equations has infinite number of solutions and solve:
x + y - z = 0,
x - 2y + z = 0,
3x + 6y - 5z = 0
Find the value of k if f(x) is continuous at $\text{x}=\frac{\pi}{2},$ where
$\text{f}\text{(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}, &\text{ x}\neq\frac{\pi}{2}\\3, &\text{ x}=\frac{\pi}{2}\end{cases}$
→$\text{f}\text{(x)}=\begin{cases}\frac{\text{k}\cos\text{x}}{\pi-2\text{x}}, &\text{ x}\neq\frac{\pi}{2}\\3, &\text{ x}=\frac{\pi}{2}\end{cases}$