Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 4 of Exercise 21.1

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

4
Find the area lying above the x - axis and under the parabola y = 4x – x2.

Given equations are:

x – axis ...... (1)

And y = 4x – x2 ...... (2)

y + 4 = – (x2 – 4x – 4) (adding 4 on both sides)

– (y + 4) = (x – 2)2

equation (2) represents a downward parabola with vertex at (2,4) and passing through (0,0) and (4,0) on the x – axis, A rough sketch is given as below: –

We have to find the area of the shaded region.

Required area

= shaded region OABO (as it is symmetrical about the x - axis)

(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 y = 4x – x2)

On integrating we get,

On applying the limits, we get,

Hence the area lying above the x - axis and under the parabola y = 4x – x2 is equal to square units.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29