Find the equation of a curve if
and the curve passes through (1,3)
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.
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
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.
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)
From these examples we can see that the definite
integral can be a positive number , a negative
number or zero.
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
Let's consider the second example
Draw the graph of y = x3 for , 1≤x≤1
The value of the
does not represent
the area below the
How can we do to calculate the shaded area?