Using integration, find the area of the region bounded by the line y – 1 = x, the x – axis and the ordinates x = – 2 and x = 3.
Given equations are:
y – 1 = x (is a line that meets at axes at (0,1) and ( – 1,0))
x = – 2 (is line parallel to y – axis at a distance of 2 units to the left)
x = 3 (is line parallel to y - axis at a distance of 3 units to the right)
A rough sketch is given as below: -
We have to find the area bounded by these three lines with the x - axis, i.e., area of the shaded region.
= shaded region ABCA + shaded region ADEA
(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 ( – 1,3) for the region ABCA and it is between ( – 2, – 1) for the region ADEA and the value of y varies)
(as y – 1 = x ⇒ y = x + 1)
(as x0 = 1)
On integrating we get,
(Combining terms with same limits)
On applying the limits, we get
Hence the area of the region bounded by the line y – 1 = x, the x – axis and the ordinates x = – 2 and x = 3 is equal to square units.