RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 17 of Exercise 21.1

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17

Sketch the graph y = |x - 5|. Evaluate . What does this value of the integral represent on the graph?

Given equations are:

y = |x - 5|


y1 = x - 5, if x - 5 ≥ 0


y1 = x - 5 …… (1), if x ≥ 5


And y2 = - (x - 5), if x - 5 < 0


y2 = - (x - 5) …… (2), if x < 5


So, equation (1) is straight line that passes thorough (5,0). Equation (2) is a line passing through (5,0) and (0,5). So, the graph of which is as follows:


15.PNG



(As when x is between (0,1) the given equation becomes y = - (x - 5) as shown in equation (2) shown ass shaded region in the above graph)


(from equation (2))



Now integrating by applying power rule, we get



Now applying the limits we get





Hence the value of represents the area of the shaded region OABC (as shown in the graph) and is equal to square units.


Chapter Exercises

Exercise 21.2
Exercise 21.3
Exercise 21.4
Exercise 21.1
MCQ

More Exercise Questions

1

Using integration, find the area of the region bounded between the line x = 2 and the parabola y2 = 8x.
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.
3

Find the area the region bounded by the parabola y2 = 4ax and the line x = a.
4

Find the area lying above the x - axis and under the parabola y = 4x – x2.
5

Draw a rough sketch to indicate the bounded between the curve y2 = 4x and the line x = 3. Also, find the area of this region
6

Make a rough sketch of the graph of the function y = 4 – x2, 0 ≤ x ≤ 2 and determine the area enclosed by the curve, the x - axis and the lines x = 0 and x = 2.
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
8

Find the area under the curve above x - axis from x = 0 to x = 2. Draw a sketch of curve also.
9

Draw the rough sketch of y2 + 1 = x, x ≤ 2. Find the area enclosed by the curve and the line x = 2
10

Draw a rough sketch of the graph of the curve and evaluate the area of the region under the curve and above the x - axis
11

Sketch the region {(x,y):9x2 + 4y2 = 36} and the find the area of the region enclosed by it, using integration
12

Draw a rough sketch of the graph of the function y=2√1–x2,x[0,1] and evaluate the are enclosed between the curve and the x–axis.
13

Determine the area under the included between the lines x = 0 and x = 1
14

Using integration, find the area of the region bounded by the line 2y = 5x + 7, x - axis the lines x = 2 and x = 8.
15

Using definite integrals, find the area of circle x2 + y2 = a2
16

Using integration, find the area of the region bounded by the following curves, after making a rough sketch: y = 1 + |x + 1|, x = - 2, x = 3, y = 0.
17

Sketch the graph y = |x - 5|. Evaluate . What does this value of the integral represent on the graph?
18

Sketch the graph y = |x + 3|. Evaluate . What does this integral represent on the graph?
19

Sketch the graph y = |x + 1|. Evaluate . What does the value of this integral represent on the graph?
20

Draw a rough sketch of the curve xy –3x – 2y – 10 = 0, x - axis and the lines x = 3, x = 4.
21

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.
22

Draw a rough sketch of the curve and find the area between x - axis, the curve and the ordinates x = 0, x = π.
23

Find the area bounded by the curve y = cosx, x - axis and the ordinates x = 0 and x = 2π.
24

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and are in the ration 2:3.
25

Compare the areas under the curves y = cos2x and y = sin2 x between x = 0 and x = π.
26

Find the area bounded by the ellipse and the ordinates x = ae and x = 0, where b2 = a2 (1 - e2) and e<1.
27

Find the area of the minor segment of the circle x2 + y2 = a2 cut off by the line .
28

Find the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2
29

Find the area enclosed by the curve x = 3 cost, y = 2 sint

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