RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 2 of Exercise 21.1

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2

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: -


2.png


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 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.


Chapter Exercises

More Exercise Questions

7

Sketch the graph of in [0,4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4