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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.
= (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)
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.