Specialist Maths: Calculus Week 6 Definite Integrals & Areas

SPECIALIST MATHS
Calculus
Week 6
Definite Integrals & Areas
Definite Integrals
• If f  x  is a continuous function for
• a  x  b and F  x    f  x  dx
• The Fundamental Theorem of Calculus
 f  x  dx  F  b   F  a 
b
a
What happened to the constant c
 f  x  dx  F  b  c   F  a   c 
b
f  x  dx  F  b   c  F  a   c
b
 a
 f  x  dx  F  b   F  a 
b
a
Definitions
 f  x  dx  F  b  F  a 
b
• The function f  x  is called the integrand

• a and b are called the lower and upper bounds
•  f  x  dx is the indefinite integral
•  a f  x  dx is called the definite integral
b
Properties of Definite Integrals
 f  x  dx  0
a
 f  x  dx    f  x  dx
b a
a b
 kf  x  dx  k  f  x  dx
b b
a a
 [ f  x   g ( x)]dx   f  x  dx   g ( x)dx
b b b
a a a
 f  x  dx   f  x  dx   g ( x)dx
b m b
a a m

Example 22 Evaluate

3
cos 2 x dx
0
Solution 22 

3
Evaluate cos 2 x dx
0


3
cos 2 x dx
0

1  3
  sin 2 x 
2 0
1 2   1 
  sin    sin 2 0  
2 3  2 
1 3 3
  0 
2 2 4
Graphics Calculator for Example 1


3
cos 2 x dx (CASSIO)
0
RUN SHIFT SETUP Arrow down to ANGLE
F2(Rad) EXIT
OPTN F4 (calc) 
F4 ( dx)
 (cos(2 X ),0,   3) Ans = 0.4330


Example 23 Evaluate in surd form :  6
cot 3 x dx
12
Solution 23


6
cot 3 x dx
12
 cos 3x
  6
dx
12 sin 3x
 1
  6
cos 3xdx
12 sin 3 x
Solution 23 (cont)
 1

6
cos 3 xdx
12 sin 3 x
u  sin 3 x
du 1 du
 3 cos 3 x cos 3 x 
dx 3 dx
 3   
u   sin
3 
 sin 
1
u ( )  sin  sin  1  12  12 4 2
6 6 2
 1 1 1 du

1
6
cos 3 xdx   1   dx
12 sin 3 x 2 u 3 dx
Solution 23 (continued)
 1
 1 1  1 
6
cos 3 xdx  ln 1  ln 
12 sin 3 x 3 3  2
1 1 1 du
1   dx 1   12 
u 3 dx  0  ln 2 
2
3  
1 1 1
 1 du 1  1
      ln 2
3 2 u 3  2
1
1  1
  ln u 
3 1  ln 2
2 6
Area Determination
• If f (x) is a positive continuous function in
the interval a  x  b then the shaded
area is given by
b y
 a
f ( x) dx y  f (x)
a b x
Area for function that is also negative
y
a b c x
b c
Area    f ( x) dx   f ( x) dx
a b
Example 24
Find the area between the curve y  4sin2x and the
x - axis for 0  x   .
Solution 24
• Find the area between the curve y  4 sin 2 x
and the x-axis for 0  x  
Step 1 draw the curve on graphics calculator and find the x intercepts
y
4 sin 2 x  0
sin 2 x  0
o 
2
 x
2 x  0  k 2 or 2 x    k 2

x  0  k or x   k
2
for 0  x  

x  0, x  or x  
2
Solution 24 continued
Step 2 state the area relation
 
Area   2
4 sin 2 x dx   4 sin 2 x dx
0
2
Step 3 calculate definite integral either with G Calc or algebraically

 
 4   4 2

  cos 2 x    cos 2 x 
 2 0  2 2
   2 cos 2 x  0 2    2 cos 2 x  
 
2
   2 cos   2 cos 0     2 cos 2  2 cos  
   2  1  2  1    2  1  2  1
 4  4  8 units 2
Area between two curves
y
y  f (x)
y  g (x)
a b y
a f ( x)  g ( x)  dx
b
Area  
Example 25
Find the area between y  cos x and y  cos 2 x
for 0  x  
Solution 25
• Find the area between
y  cos x and y  cos 2 x for 0  x  
Step 1 draw the graph.
y
y  cos 2 x
y  cos x
Step 2 find the points of intersection
cos x  cos 2 x
cos x  2 cos 2 x  1
0  2 cos 2 x  cos x  1
0  (2 cos x  1)(cos x  1)
1
cos x   or cos x  1
2
2
x or x  0
3
Step 3 state the definite integral for the area and calculate it
2
Area  
0
3
 cos x  cos 2 x  dx
2
 1  3
 sin x  sin 2 x 
 2 0
 2 1 4   1 
  sin  sin    sin 0  sin 0 
 3 2 3   2 
3 1  3
 
  
2 2  2 
3 3
 
2 4
2 3 3
3 3
  units 2
4 4
This Week
• Text Book Pages 258 to 263
• Exercise 7D3 Q 1 – 2
• Questions 5 & 6 from Review Sets 6A – 6C
• Review Sets 7A – 7D

Specialist Maths: Calculus Week 6 Definite Integrals & Areas

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SPECIALIST MATHS

• The Fundamental Theorem of Calculus

• The function f  x  is called the integrand

RUN SHIFT SETUP Arrow down to ANGLE

 (cos(2 X ),0,   3) Ans = 0.4330

Step 3 calculate definite integral either with G Calc or algebraically

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