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JoshuaWong902269

integration

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Example 1
Find the area A under
y = x over the interval
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By integration,
Δx
2
0
2
2
2
2
0
0
2
b
b
x
dx
x
b
b

















Area of
triangle,
Area = ½ (base)
x (height)
Area = ½ (b)(b)
= 2
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 
2
5
2
3
-
4(3)
3
1
2
4
3
1
x
-
4
3
1
)
(
1
Average
2
3
0
2
3
0





















x
x
dx
dx
x
f
a
b
b
a
Example 2

Example 3
Calculate the area bounded by
the x-axis and the parabola
2
6 x
x
y 



Solution:
2
3
2
3
3
2
2
3
2
6
)
6
(


 










x
x
x
dx
x
x



















3
27
2
9
18
3
8
2
12
6
5
20


Example 4
2
1
2
)
(
of
graph
the
and
axis
-
between x
region
the
of
area
the
Find
2
3






x
x
x
x
x
f

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12
5
3
4
)
2
(
:
0
1
2
3
4
0
1
2
3















x
x
x
dx
x
x
x
Solution
3
8
3
4
)
2
(
2
0
2
3
4
2
0
2
3














x
x
x
dx
x
x
x
2
units
12
37
3
8
12
5
area
enclosed
Total 




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Find the solution by 3 steps:
1. Sketch the curves and shade the required
area.
2. Find the limits of integration by solving
simultaneous equations.
3. Calculate the area.
Find the area enclosed by the parabola y=2 – x2
and the line y = – x.
Example 5

origin
through
passing
slope
negative
line
straight
axis
-
y
on
intercept
c
slope,
m
where
,
line
straight
of
Equation
:
Revision









x
y
c
mx
y
y
x
y = – x

2
,
0
when
:
axis
-
on x
on
intersecti
Find
2
,
0
when
:
axis
-
y
on
on
intersecti
Find
2 2







x
y
y
x
x
y
y
x

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(minima)
positive
(maxima),
negative
when
:
Note
(maxima)
2
(0,2)
point
turning
2
,
0
when
0
,
0
when
(slope)
2
2
2
2
2
2
2













dx
y
d
dx
y
d
y
x
x
dx
dy
x
dx
dy
x
y
y
x

)
2
(2,
and
1)
,
1
(
are
points
on
Intersecti
2
2
When
1
1
When
2
,
1
0
)
2
)(
1
(
0
2
2
:
and
2
for
Solve
:
curves
2
between
points
on
intersecti
Finding
2
2
2
























, y
x
, y
x
x
x
x
x
x
x
x
x
x
y
x
y

Combining both
graphs,
Note:
- Slope of curves
- Turning point
- Maxima
- Intersection points

2
2
1
3
2
2
1
2
2
1
2
unit
2
9
3
1
2
1
2
3
8
2
4
4
3
2
2
)
2
(
)]
(
)
2
[(
)]
(
)
(
[












































x
x
x
dx
x
x
dx
x
x
dx
x
g
x
f
A
b
a
Solution for
area between
2 curves:

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Solution:
Area A
Limits of integration are a=0, b=2.
Area B
Solving simultaneously 2
and 

 x
y
x
y
4
,
1
0
)
4
)(
1
(
0
4
5
4
4
)
2
(
2
2
2













x
x
x
x
x
x
x
x
x
x

3
10
)
2
(
area
Total
4
2
2
0




 
 dx
x
x
dx
x
2)
-
(x
-
)
(
)
(
:
4
2
For
)
(
)
(
:
2
0
For
4
limit
hand
Right
x
x
g
x
f
x
x
x
g
x
f
x










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Solution:
Solving simultaneous equation of
gives roots y=-1, y=2.
Limits a=0,b=2.
2
and
2


 y
x
y
x
3
10
)
2
(
)]
(
)
(
[
2
0
2








dy
y
y
dy
y
g
y
f
A
b
a

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The process of finding area of some plane region is called Quadrature. In this chapter we shall find the area bounded by some simple plane curves with the help of definite integral. For solving (i) The area bounded by a cartesian curve y = f(x), x-axis and ordinates x = a and x = b is given by, the problems on quadrature easily, if possible first draw the rough sketch of the required area. b Area =  y dx a b =  f(x) dx a In chapter function, we have seen graphs of some simple elementary curves. Here we introduce some essential steps for curve tracing which will enable us to determine the required area. (i) Symmetry The curve f(x, y) = 0 is symmetrical  about x-axis if all terms of y contain even powers.  about y-axis if all terms of x contain even powers.  about the origin if f (– x, – y) = f (x, y). Examples based on Area Bounded by A Curve For example, y2 = 4ax is symmetrical about x-axis, and x2 = 4ay is symmetrical about y- axis and the curve y = x3 is symmetrical about the origin. (ii) Origin Ex.1 Find the area bounded by the curve y = x3, x-axis and ordinates x = 1 and x = 2. 2 Sol. Required Area = ydx 1 If the equation of the curve contains no constant 2 = x3 dx = Lx4 O2 15 MN PQ= Ans. term then it passes through the origin. For example x2 + y2 + 2ax = 0 passes through origin. (iii) Points of intersection with the axes If we get real values of x on putting y = 0 in the equation of the curve, then real values of x and y = 0 give those points where the curve cuts the x-axis. Similarly by putting x = 0, we can get the points of intersection of the curve and y-axis. 1 4 1 4 Ex.2 Find the area bounded by the curve y = sec2x,  x-axis and the line x = 4 /4 Sol. Required Area = ydx x0 For example, the curve x2/a2 + y2 /b2 = 1 intersects the axes at points (± a, 0) and (0, ± b) . (iv) Region /4 = sec2 0 xdx = tan x /4 = 1 Ans. Write the given equation as y = f(x) , and find minimum and maximum values of x which determine the region of the curve. For example for the curve xy2 = a2 (a – x) Ex.3 Find the area bounded by the curve y = mx, x-axis and ordinates x = 1 and x = 2 2 Sol. Required area = y dx . 1 y = a a  x x 2 = mxdx = LMmx2 2 Now y is real, if 0 < x  a , so its region lies between the lines x = 0 and x = a. 1 MN2 = m ( 4 – 1) = 2 PQ1 FGH3 JKm Ans. Ex.4 Find the area bounded by the curve y = x (1– x)2 and x-axis. Sol. Clearly the given curve meets the x-axis at (0,0) and (1,0) and for x = 0 to 1, y is positive Ex.7 Find the area bounded between the curve y2 = 2y – x and y-axis. Sol. The area between the given curve x = 2y – y2 and y-axis will be as shown in diagram. so required area- Y = xb1 xg2 dx 0 1 = ex  2x2  x3 jdx 0 O (0,0) (1,0) X Lx2 2x3 x4 O1 1 2 1 1 = M  P= – + = MN2 3 4 PQ0 2 3 4 12 Ans.  Required Area = e2y  y2 jdy (i

Integration

suefee

The document discusses integration and indefinite integrals. It covers determining integrals by reversing differentiation, integrating algebraic expressions like constants, variables, and polynomials. It also discusses determining the constant of integration and using integration to find equations of curves from their gradients. Examples are provided to illustrate integrating functions and finding volumes generated by rotating an area about an axis.

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1. This document provides instructions for completing Section A of a mathematics exam. Students must use a pencil and fill in answers on an answer sheet by making a horizontal line in the correct space. 2. The answer sheet contains the student's name, date of birth, and other identifying information. Students should check this is correct and report any errors. 3. There is one correct answer for each question, and rough work should not be done on the answer sheet.

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