1. Integrate:

2. Find the equation of a curve if
and the curve passes through (1,3)

3. The rate of change in pressure, p units, at a depth
x cm from the surface of a liquid is given by
p'(x) = 0.03 x2. If the pressure at the surface is 10
units, find the pressure at a depth of 5 cm.
11.25 units

4. The growth rate of a city's population has been
modelled by the equation:
where t is the time in years after 1995 and N is the
population size. In the year 2000 the population
numbered 32 000.What will the population be in
2010?
76981

5. Definite Integrals
By the end of the lesson you will be able to:
• Calculate definite integrals.
• Apply properties of definite integrals.
• Understand the relationship between definite
integrals and area.

6. The definite integral
If f(x) is a continous function in the interval [a,b],
and F(x) is any antiderivative, then:
is called the definite integral
and it is equal to : F(b) ­ F(a)

7. upper limit
lower limit

8. From these examples we can see that the definite
integral can be a positive number , a negative
number or zero.

9. Let's consider our first example:
the function in this case is
f(x) = x
Let's draw f(x) between 0 and 3.
In this case the value of the definite integral
represents the area below the curve, between 0
and 3.

10. Let's consider the second example
Draw the graph of y = x3 for , ­1≤x≤1
The value of the
definite integral
does not represent
the area below the
curve.
How can we do to calculate the shaded area?

11. Shaded area =

12. Let's consider now the third example:
and the graph of y = 2x+1 for ­3≤x≤1
Again the value of the
definite integral does
not represent the area
below the curve.
How can we calculate
this area?

14. Conclusions:
•
may be positive, negative or zero.
• When f(x) > 0 in [a , b ] then
represents the area under the curve y = f(x), above
the x­axis in the interval [a , b ].

15. Properties of the definite integral
1.
2.
3.
where a≤c≤b

16. 4.
k ∈ lR
5.

17. Calculate the area under the curve y=x2 betwen
x=1 and x=3.

18. Calculate:
Calculate the area beneath the curve y =x (x ­ 1)
from x= 0 to x =1
check graph first !

19. Find the area between the curve y = x2 + 4 x and the x­ axis
from x = ­2 to x = 0 .
check graph first !

20. Find the area between the curve y = x2 + 4 x and the x­ axis
from x = ­2 to x = 2 .
16
Check graph first !!

21. Use of GDC:
Menu Run:
OPTN
CALC (F4)
(F4)
upper limit
function
lower limit

22. Menu Graph:
Y1 =x2
DRAW
GSOLV (F5)
(F3)
Enter the lower limit and press EXE
Enter the upper limit and press EXE

23. Book page 405 Ex 13G :
1a) 2a) and 4, 5, 6.
Page 411 Ex 13H: 1 to 5