Using integration, find the area of the region bounded by the line $2y = 5x + 7$, X-axis and the lines $x = 2$ and $x = 8$.

$96$ square units

The correct answer is Option (2) → $96$ square units

We have,

$2y = 5x + 7$

$\Rightarrow y = \frac{5x}{2} + \frac{7}{2}$

$∴\text{Area of shaded region} = \frac{1}{2} \int\limits_{2}^{8} (5x + 7) \, dx = \frac{1}{2} \left[ 5 \cdot \frac{x^2}{2} + 7x \right]_{2}^{8}$

$= \frac{1}{2} [5 \cdot 32 + 7 \cdot 8 - 10 - 14] = \frac{1}{2} [160 + 56 - 24]$

$= \frac{192}{2} = 96 \text{ sq. units}$