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STUDY PACKAGE
Subject : Mathematics
Topic : Area Under Curve (Quadrature)
Available Online : www.MathsBySuhag.com

Index
1. Theory
2. Short Revision
3. Exercise (Ex. 1 + 5 = 6)
4. Assertion & Reason
5. Que. from Compt. Exams
6. 39 Yrs. Que. from IIT-JEE(Advanced)
7. 15 Yrs. Que. from AIEEE (JEE Main)
Student’s Name :______________________
Class :______________________
Roll No. :______________________

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Area Under Curve

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page 2 of 21
1. Curve Tracing :
To find the approximate shape of a curve, the following procedure is adopted in order:
(a) Symmetry:
(i) Symmetry about x  axis:
If all the powers of ' y ' in the equation are even then the curve is symmetrical about the x  axis.

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E.g.: y2 = 4 a x.
(ii) Symmetry about y  axis:
If all the powers of ' x ' in the equation are even then the curve is symmetrical about the y  axis.

E.g.: x2 = 4 a y.
(iii) Symmetry about both axis;
If all the powers of ' x ' and ' y ' in the equation are even, the curve is symmetrical about the axis of ' x
' as well as ' y '.

E.g.: x2 + y2 = a2.
(iv) Symmetry about the line y = x:
If the equation of the curve remains unchanged on interchanging ' x ' and ' y ', then the curve is
symmetrical about the line y = x.

E.g.: x3 + y3 = 3 a x y.

(v) Symmetry in opposite quadrants:

If the equation of the curve remains unaltered when ' x ' and ' y ' are replaced by  x and  y respectively,
then there is symmetry in opposite quadrants.

E.g.: x y = c2.

(b) Find the points where the curve crosses the xaxis and also the yaxis.

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dy
(c) Find dx and equate it to zero to find the points on the curve where you have horizontal tangents.
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(d) Examine if possible the intervals when f (x) is increasing or decreasing.

page 3 of 21
(e) Examine what happens to ‘y’ when x   or x   .

(f) Asymptotes :
Asymptoto(s) is (are) line (s) whose distance from the curve tends to zero as point on curve moves towards
infinity along branch of curve.
(i) If Lt f(x) =  or Lt f(x) = – , then x = a is asymptote of y = f(x)
x a x a

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(ii) If Lt f(x) = k or Lt f(x) = k, then y = k is asymptote of y = f(x)
x   x  

f(x)
(iii) If x Lt
 = m1, x Lt
  (f(x) – m 1x) = c, then y = m 1x + c1 is an asymptote. (inclined to right)
x
f(x)
(iv) If Lt = m2, Lt (f(x) – m2x) = c2, then y = m2x + c2 is an asymptote (inclined to left)
x   x x  
Example : Find asymptote of y = e–x
Solution. Lim y = Lim e–x = 0
x  x 

 y = 0 is asymptote.

Example : Find asymptotes of xy = 1 and draw graph.

1
Solution y=
x
Lim y = Lim 1 =   x = 0 is asymptote.
x 0 x 0 x

Lim y = Lim 1 = 0  y = 0 is asymptote.

x  x  x

1
Example : Find asymptotes of y = x + and sketch the curve.
x
1
Solution Lim y = Lim  x   = + or –
x 0 x 0  x
 x = 0 is asymptote.
1
Lim y = Lim  x   = 
x 0 x 0  x
 there is no asymptote of the type y = k.
1 
Lim y = Lim 1   =1
x  x x   x2 
1
Lim (y – x) = Lim  x   x  = Lim 1 = 0
x  x   x  x  x

 y = x + 0  y = x is asymptote.
A rough sketch is as follows

2. Quadrature :
b
(a) If f(x)  0 for x  [a, b], then area bounded by curve y = f(x), x-axis, x-axis, x = a and x = b is  f ( x) dx
a

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page 4 of 21
Example : Find area bounded by the curve y = n x + tan–1x and x-axis between ordinates x = 1 and x = 2.

Solution y = n x + tan–1x
dy 1 1
Domain x > 0 = + >0
dx x 1 x2
It is increasing function

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Lt y = Lt (n x + tan–1x) = 
x  x 

Lt Lt  (n x + tan–1x) = 

y = x 0
x 0 
A rought sketch is as follows
2
1
 Required area = x ) dx
 (n x  tan
1
2
 1 1 2 
=  x n x  x  x tan x  n (1  x )
 2 1
1 1
= 2 n 2 – 2 + 2 tan–12 – n 5 – 0 + 1 – tan–1 1 + n 2
2 2
5 1 
= n 2 – n 5 + 2 tan–12 – –1
2 2 4

(b) If f(x)  0 for x  [a, b], then area bounded by curve y = f(x), x-axis, x = a and x = b is –  f ( x) dx
a

Example : Find area bounded by y = log 1 x and x-axis between x = 1 and x = 2.

2

Solution. A rought sketch of y = log 1 x is as follows

2
2 2
Area =– 
1
log 1 x dx = –
2
 log e x . log 1 e dx
2
1

= – log 1 e . [ x loge x  x]12

2

= – log 1 e . (2 loge2 – 2 – 0 + 1)
2

= – log 1 e . (2 loge 2 – 1)
2
Note : If y = f(x) does not change sign an [a, b], then area bounded by y = f(x), x-axis between
b

ordinates x = a, x = b is  f (x) dx
a
.

(c) If f(x) > 0 for x  [a,c] and f(x) < 0 for x  [c,b] (a < c < b) then area bounded by curve y = f(x) and x–axis
c b
between x = a and x = b is
 f (x) dx   f (x) dx .
a c
Example : Find the area bounded by y = x3 and x–axis between ordinates x = – 1 and x = 1.
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0 1

Solution Required area =   x 3 dx  x 3 dx


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1 0
0 1
x4  x3 
=   + 4

page 5 of 21
4  
1 0
 1 1 1
= 0 –   + –0=
 4 4 2
b


Note : Area bounded by curve y = f(x) and x–axis between ordinates x = a and x = b is | f ( x ) | dx .
a

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(d) If f(x) > g(x) for x[a,b] then area bounded by curves y = f(x) and y = g(x) between ordinates x = a and
b

x = b is  f (x)  g(x)dx .
a

Example : Find the area enclosed by curve y = x2 + x + 1 and its tangent at (1,3) between ordinates x = – 1 and
x = 1.
dy
Solution. = 2x + 1
dx
dy
= 3 at x = 1
dx
Equation of tangent is
y – 3 = 3 (x – 1)
y = 3x
1
2
Required area =  (x
1
 x  1  3 x ) dx

1 1
2 x3 
= 
( x  2 x  1) dx   x 2  x
3  1
1

1   1 
=   1  1 –    1  1
 3   3 
2 8
=
+2=
3 3
Note : Area bounded by curves y = f(x) and y = g (x) between ordinates x = a and x = b is
b

 | f (x)  g(x) | dx .
a
(e) If g (y)  0 for y  [c,d] then area bounded by curve x = g(y) and y–axis between abscissa y = c and
d
y = d is
 g(y)dy
y c

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
Example : Find area bounded between y = sin–1x and y–axis between y = 0 and y = .
2
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Solution y = sin–1 x
 x = sin y


page 6 of 21
2
Required area =  sin y dy
0


=  cos y 
2
0
= – (0 – 1) = 1
Note : The area in above example can also evaluated by integration with respect to x.

Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : 0 903 903 7779, 0 98930 58881.

Area = (area of rectangle formed by x = 0, y = 0 , x = 1, y = ) – (area bounded by y = sin–1x,
2
x–axis between x = 0 and x = 1)
1
 1 
=
2
×1–  sin
0
x dx =

– (x sin–1x + 1 x 2 )
1

  
= –   0  0  1 = 1
2 2 
Some more solved examples
Example : Find the area contained between the two arms of curves (y – x)2 = x3 between x = 0 and x = 1.

Solution (y – x)2 = x3  y = x ± x 3/2

For arm
dy 3 1/2
y = x + x3/2  =1+ x >0 x > 0.
dx 2
y is increasing function.
For arm
dy 3 1/2
y = x – x3/2  =1– x
dx 2
1
dy 4 d2 y 3  4
=0 x= , 2
  x 2 < 0 at x =
dx 9 dx 4 9
4
 at x = y = x – x3/2 has maxima.
9
1
3/2
Required are a =  (x  x  x  x 3 / 2 ) dx
0

1 1
2 x5 / 2  4
=2  x 3 / 2 dx   =
5 / 2  5
0 0

Example : Find area contained by ellipse 2x2 + 6xy + 5y2 = 1

Solution. 5y2 + 6xy + 2x2 – 1 = 0

 6 x  36 x 2  20(2x 2  1)
y=
10

 3x  5  x 2
y=
5
 y is real  R.H.S. is also real.
 – 5 <x< 5
If x=– 5 , y=3 5
If x= 5, y = –3 5
1
If x = 0, y=+
5

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1
If y = 0, x=+
2
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5  3 x  5  x 2  3x  5  x 2 
 
Required area =  5

5  dx

page 7 of 21
 5 
5
2 5  x 2 dx
=
5

 5

5
4
 5  x 2 dx

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=
5 0

Put x = 5 sin  : dx = 5 cos  d

L.L : x = 0  = 0

U.L : x = 5   =
2

2
4
=  5  5 sin2  5 cos d
5
0


2
2 1 
=4  cos
0
d = 4
2 2
=

Example : Let A (m) be area bounded by parabola y = x2 + 2x – 3 and the line y = mx + 1. Find the least area
A(m).
Solution. Solving we obtain
x2 + (2 – m) x – 4 = 0
Let  be roots  = m – 2,  = – 4

2
A (m) =  (mx 1 x

 2x  3) dx


2
=  ( x

 (m  2) x  4) dx


 x3 x2 
  ( m  2)  4x 
=  3 2 
 

3  3 m  2 2
=  (   2 )  4 (   )
3 2

1 2 (m  2)
= | – |.  (     2 )  (   )  4
3 2
1 (m  2)
=
2

(m  2)2  16  3 (m  2)  4  2 (m  2)  4 
1 8
= (m  2)2  16 (m  2)2 
6 3
1
A(m) =  2
6 (m  2)  16
3/2

1 32
Leas A(m) = (16)3/2 = .
6 3

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Self Practice Problems

1. Find the area between curve y = x2 – 3x + 2 and x–axis

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1
(i) bounded between x = 1 and x = 2. Ans.
6

page 8 of 21
(ii) bound between x = 0 and x = 2. Ans. 1

2. Find the area included between curves y = 2x – x2 and y + 3 = 0.

32
Ans.
3

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3. Find area between curves y = x2 and y = 3x – 2 from x = 0 to x = 2.
Ans. 1

4. Curves y = sinx and y = cosx intersect at infinite number of points forming regions of equal area between
them calculate area of one such region.
Ans. 2 2

5. Find the area of the region bounded by the parabola (y – 2)2 = (x – 1) and the tangent to it at ordinate y = 3
and x–axis.
Ans. 9

6. Find the area included between y = tan–1x, y = cot–1x and y–axis.

Ans. n2

7. Find area common to circle x2 + y2 = 2 and the parabola y2 = x.

 3 2
Ans.  
3 2 3

4  x2
8. Find the area included between curves y = and 5y = 3|x| – 6.
4  x2
8
Ans. 2 –
5

1
9. Find the area bounded by the curve |y| + = e–|x|.
2
Ans. 2 (1–n2)

10. Find the area of loop y2 = x (x – 1)2.

8
Ans.
15

11. Find the area enclosed by |x| + |y| < 3 and xy > 2.
Ans. 3–4n2

12. Find are bounded by x2 + y2 < 2ax and y2 > ax, x > 0.
 3  8 
Ans.   a2.
 6 

For 39 Years Que. from IIT-JEE(Advanced) &

15 Years Que. from AIEEE (JEE Main)
we distributed a book in class room

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