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APPLICATION OF INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF INTEGRALS4 Marks
Question
Using integration, find the area of the triangular region whose sides have the equations y = 2x + 1, y = 3x + 1 and x = 4.
✓

Answer

Equations of one side of triangle is

y = 2x + 1 ...(i)

second line of triangle is y = 3x + 1 ...(ii)

third line of triangle is x = 4 ...(iii)

Solving eq. (i) and (ii), we get x = 0 and y = 1

$\therefore$ Point of intersection of lines (i) and (ii) is A(0, 1)

Putting x = 4 in eq. (i), we get y = 9

$\therefore$ Point of intersection of lines (i) and (iii) is B(4, 9)

Putting x = 4 in eq. (i), we get y = 13

$\therefore$ Point of intersection of lines (ii) and (iii) is C(4, 13)

$\therefore$ Area between line (ii) i.e., AC and x-axis

$=\begin{vmatrix}\int\limits^4_{0}\text{y dx} \end{vmatrix}=\begin{vmatrix}\int\limits^4_{0}(3\text{x}+1)\text{dx} \end{vmatrix}=\bigg(\frac{3\text{x}^2}{2}+\text{x}\bigg)^4_0$

= 24 + 4 = 28 sq. units ...(iv)

Again Area between line (i) i.e., AB and x-axis

$=\begin{vmatrix}\int\limits^4_{0}\text{y dx} \end{vmatrix}=\begin{vmatrix}\int\limits^4_{0}(2\text{x}+1)\text{dx} \end{vmatrix}=\bigg(\text{x}^2+\text{x}\bigg)^4_0$

= 16 + 4 = 20 sq. units …(v)

Therefore, Required area of $\Delta\text{ABC}$

= Area given by (iv) - Area given by (v)

= 28 - 20 = 8 sq. units

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