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.

Required area

= 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 x^{0} = 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.

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