RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 7 of Exercise 21.1

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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

Given equations are:

x – axis ...... (1)


x = 0 ...... (2)


x = 4 ...... (3)


And , 0 x 4 ...... (4)


equation (4) represents a half parabola with vertex at ( – 1,0) and passing through (4,0) on x – axis, equation (3) represents a line parallel to y - axis at a distance of 4 units and equation (2) represents y - axis.


A rough sketch is given as below: -


6.PNG


We have to find the area of the shaded region.


Required area


= shaded region AOBC


(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 (0,4) and the value of y varies)


(as )



Substitute


So the above equation becomes,



On integrating we get,



On applying the limits we get,





Hence the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4 is equal to square units.


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Chapter Exercises

More Exercise Questions

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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