RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 13 of Exercise 21.1

Listen NCERT Audio Books to boost your productivity and retention power by 2X.

13

Determine the area under the included between the lines x = 0 and x = 1

Given equations are :

...... (1)



x = 0 (y – axis)


x = 1 (represents a line parallel to y - axis at a distance 1 to the right)


equation (1) represents a half eclipse that is symmetrical about the x - axis and also about the y - axis with center at origin.


A rough sketch is given as below: -


11.PNG


We have to find the area of shaded region.


Required area


= (shaded region OABCO)


(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 (0,1) and the value of y varies)


(as )



Substitute


So the above equation becomes,




We know,


So the above equation becomes,




Apply reduction formula:



On integrating we get,




Undo the substituting, we get





On applying the limits we get,





Hence the area under the included between the lines x = 0 and x = 1 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