
Mathematics 1

səhifə  1/4  tarix  26.10.2023  ölçüsü  2,11 Mb.   #131174 
 MES2 4th week MES 24th week  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  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 xaxis.  If f(x) ≤ 0 :
 =“” area between f(x) and
the xaxis.  If a function crosses the xaxis
we have to split the region/ above and all below and work out the areas for each part separately. Curves above and below the xaxisexample1.49  What is the area bounded by the graph of y = (x−1)(x−2) and the coordinate axes?
 Solution:
 y = (x−1)(x−2)=x23x+2
 y changes sign at x=1
and x=2 Area A: =(+2)=+2== Area B:=(+2)=(+4)== Total area enclosed: +=1  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 )
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