Unformatted text preview: = F(b) − F(a)
over [a, b]
over [a, b]
In both SIGN and MAGNITUDE as we saw in the examples 1 and 2.
2. If f(x) is negative (below the xaxis) then :
negative
AREA under f(x)
=
over [a, b] [ F(b) − F(a) ] when the direction of integration
direction
over [a, b] is positive
positive In both SIGN and MAGNITUDE as we saw in the examples 3 and 5.
AREA under f(x)
=
over [a, b] direction
− [ F(b) − F(a) ] when the direction of integration
over [a, b] is negative
negative In both SIGN and MAGNITUDE as we saw in the examples 4 and 6. 155 continuous
ov
may
ind
We may use the AREA under continuous f(x) ov er [a, b] to ffind the
CHANGE in the INTEGRAL F(x) over [a, b] in the positive direction
as follows.
positiv
1. We must break up the AREA under f(x) over [a, b] into segments of positive
positi
areas and negative areas as we did in example 7.
negative
positive area = area above the xaxis
negative area = area below the xaxis
Here we implicitly assumed that the direction of integration is positive .
direction
positive neg tiv
positiv
2. We SUM UP all the negative and positive areas to get a net result, say A . Then:
ne
positi
CHANGE in F(x) over [a, b] in the positive direction = F(b) − F(a) = A
positive EXERCISES
The exercises are to show the relationship between the area under the
area
cur v e f(x) = F’(x) and the CHANGE IN VALUE of the Integral F(x).
Exercise 1: Repeat examples 7 and 8 with f(x) =  2x
Ex er cise 2: Integrate f(x) =  2x over the inter val [ 2,  1] from LEFT to
Exer
ercise
2 f rom  2 to  1. What
RIGHT. Verify it against the CHANGE in F(x) =  x
is the area under the curve ?
Exer
ercise
Ex er cise 3: Integrate f(x) =  2x over the interval [ 2,  1] from RIGHT
2 f rom  1 to  2. What
to LEFT. Verify it against the CHANGE in F(x) =  x
is the area under the curve ?
Exercise 4: What is the CHANGE in the value of the function sin (x) over
the inter vals [0, π /2], [ π /2, 3π /2], [ π , 2 π ], [0, 2 π ] . Use the fact that
the INTEGRAL sin (x) is the ANTIDERIVATIVE of cos (x). Draw the curve of
cos (x) over [0, 2π ]. For each interval verify that the CHANGE IN AREA is
equal to the CHANGE IN VALUE of sin (x) over the interval going LEFT to
RIGHT. Repeat the exercise going RIGHT to LEFT.
156 30. Area under f(x) and Plottin...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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