Mathematics 1

MES 2-4th week

Definite Integrals

• As you remember the second way of think of integrals is as of the area under a curve (or to be more precise between the curve and x –axes) which leads to the idea of a definite integral, one with limits.
• Area=.
• It is easy, if the area

is rectangle

If f(x) is given by the

following function:

• What is ?
• Solution:

192+106+192=136

•  

Curves above and below the x-axis

• If the curve is below the line the integral will be a negative number. To get the area (which must be positive) we must change the sign.
• Hence if f(x) ≥ 0 :
• = area between f(x) and

the x-axis.

• If f(x) ≤ 0 :
• =“-” area between f(x) and

the x-axis.

• If a function crosses the x-axis

we have to split the region/

integral into parts that are all

above and all below and work out the areas for each part separately.

•  

Curves above and below the x-axis-example1.49

• What is the area bounded by the graph of y = (x−1)(x−2) and the coordinate axes?
• Solution:
• y = (x−1)(x−2)=x2-3x+2
• y changes sign at x=1

and x=2

Area A:

=(-+2)=-+2==

Area B:=-(-+2)=(-+-4)==

Total area enclosed: +=1

•  

Area between two graphs

• Sometimes we wish to find the area enclosed between two graphs. We can do this by subtracting the area of a smaller region from the area of a large region.
• To determine the areas we need to determine the relevant interval for x to integrate over. To do this we must determine where the graphs intersect.

Area between two graphs - example1.51

• What is the area bounded by the graphs of y = x2 and y = x3, and the lines x = 0 and x = 2?

Solution:

In the Area A )