ib maths sl definite integrals
TRANSCRIPT
Integrate:
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.
11.25 units
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
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.
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)
upper limit
lower limit
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:
f(x) = xthe function in this case is
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.
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?
Shaded area =
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?
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 ].
•
•
Properties of the definite integral
1.
2.
3.where a≤c≤b
4.k ∈ lR
5.
Calculate the area under the curve y=x2 betwen x=1 and x=3.
Calculate the area beneath the curve y =x (x 1) from x= 0 to x =1
Calculate:
check graph first !
Find the area between the curve y = x2 + 4 x and the x axis from x = 2 to x = 0 .
check graph first !
Find the area between the curve y = x2 + 4 x and the x axis from x = 2 to x = 2 .
16
Check graph first !!
Use of GDC:
Menu Run: OPTN
CALC (F4)
(F4)
functionlower limit
upper limit
Menu Graph:
Y1 =x2
DRAW
GSOLV (F5)
(F3)
Enter the lower limit and press EXE
Enter the upper limit and press EXE
Book page 405 Ex 13G : 1a) 2a) and 4, 5, 6.
Page 411 Ex 13H: 1 to 5