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
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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