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

Given equations are:

2y = 5x + 7 ...... (1)

x = 2 ...... (2)

x = 8 ...... (3)

Equation (1) represents line passing through and . Equation (2), (3) shows line parallel to y - axis passing through (2,0), (8,0) respectively.

A rough sketch of curves is as below:

We have to find the area of shaded region.

Required area

= (shaded region ABCDA)

(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)

(As x is between (2,8) and the value of y varies)

(as )

Now integrating by applying power rule, we get

Now applying the limits we get

Hence the area of the region bounded by the line 2y = 5x + 7, x - axis the lines x = 2 and x = 8 is equal to 96 square units.