Calculus 5.2

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5.2 Definite Integrals
Greg Kelly, Hanford High School, Richland,
Washington
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When we find the area under a curve by adding
rectangles, the answer is called a Rieman sum.
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
subinterval
partition
Subintervals do not all have to be the same size.
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If we use subintervals of equal length, then the
length of a subinterval is
The definite integral is then given by
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Leibnitz introduced a simpler notation for the
definite integral
Note that the very small change in x becomes dx.
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upper limit of integration
Integration Symbol
integrand
variable of integration (dummy variable)
lower limit of integration
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We have the notation for integration, but we
still need to learn how to evaluate the integral.
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In section 5.1, we considered an object moving at
a constant rate of 3 ft/sec.
Since rate . time distance
If we draw a graph of the velocity, the distance
that the object travels is equal to the area
under the line.
After 4 seconds, the object has gone 12 feet.
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If the velocity varies
Distance
(C0 since s0 at t0)
After 4 seconds
The distance is still equal to the area under the
curve!
Notice that the area is a trapezoid.
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What if
We could split the area under the curve into a
lot of thin trapezoids, and each trapezoid would
behave like the large one in the previous example.
It seems reasonable that the distance will equal
the area under the curve.
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We can use anti-derivatives to find the area
under a curve!
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Lets look at it another way
Then
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The area of a rectangle drawn under the curve
would be less than the actual area under the
curve.
The area of a rectangle drawn above the curve
would be more than the actual area under the
curve.
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As h gets smaller, min f and max f get closer
together.
This is the definition of derivative!
initial value
Take the anti-derivative of both sides to find an
explicit formula for area.
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As h gets smaller, min f and max f get closer
together.
Area under curve from a to x antiderivative at
x minus antiderivative at a.
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Area
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Example
Find the area under the curve from x1 to x2.
Area from x0 to x1
Area from x0 to x2
Area under the curve from x1 to x2.
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Example
Find the area under the curve from x1 to x2.
To do the same problem on the TI-89
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Example
Find the area between the x-axis and the
curve from to .
pos.
neg.
On the TI-89
If you use the absolute value function, you dont
need to find the roots.
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