Question

Solve the following :

Find the area of the region bounded by the straight line 2y = 5x + 7, X-axis and x = 2, x = 5.

Answer 1

The equation of the line is 2y = 5x + 7,

i.e. y = ${\displaystyle \frac{5}{2}x+\frac{7}{2}}$

Required area = area of the region ABCDA

= area under the line y = ${\displaystyle \frac{5}{2}x+\frac{7}{2}}$ between x = 2 and x = 5

= ${\displaystyle {\int }_2^5(\frac{5}{2}x+\frac{7}{2})\cdot dx}$

= ${\displaystyle \frac{5}{2}\cdot {\int }_2^{5}x\cdot dx+\frac{7}{2}{\int }_2^{5}1\cdot dx}$

= ${\displaystyle \frac{5}{2}{\left[\frac{x^2}{2}\right]}_2^5+\frac{7}{2}{\left[x\right]}_2^5}$

= ${\displaystyle \frac{5}{2}[\frac{25}{2}-\frac{4}{2}]+\frac{7}{2}[5-2]}$

= ${\displaystyle \frac{5}{2}\times \frac{21}{2}+\frac{21}{2}}$

= ${\displaystyle \frac{105}{4}+\frac{42}{4}}$

= ${\displaystyle \frac{147}{4}\text{sq units}}$.