The area of the region bounded by the line \(2y = 5x + 7\), \(x\)-axis and the lines \(x = 1\) and \(x = 3\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 17
The line is
\( \displaystyle y = \frac{5x + 7}{2} \)
For \(1 \le x \le 3\), y is positive, so the region lies above the x-axis and the area under the line is
\( \displaystyle \int_{1}^{3} \frac{5x + 7}{2}\,dx = \frac{1}{2}\Big[\frac{5x^2}{2} + 7x\Big]_{1}^{3} \)
\( \displaystyle = \frac{1}{2}\left[\left(\frac{45}{2} + 21\right) - \left(\frac{5}{2} + 7\right)\right] \)
\( \displaystyle = \frac{1}{2}\left(20 + 14\right) = 17 \)
Shortcut: the region is a trapezium with parallel sides y(1) = 6 and y(3) = 11 and width 3 − 1 = 2:
\( \displaystyle \text{Area} = \frac{1}{2}(6 + 11) \times 2 = 17 \)