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5.2 Definite Integrals

Greg Kelly, Hanford High School, Richland,

Washington

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.

(No Transcript)

If we use subintervals of equal length, then the

length of a subinterval is

The definite integral is then given by

Leibnitz introduced a simpler notation for the

definite integral

Note that the very small change in x becomes dx.

upper limit of integration

Integration Symbol

integrand

variable of integration (dummy variable)

lower limit of integration

We have the notation for integration, but we

still need to learn how to evaluate the integral.

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.

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.

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.

We can use anti-derivatives to find the area

under a curve!

Lets look at it another way

Then

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.

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.

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.

Area

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.

Example

Find the area under the curve from x1 to x2.

To do the same problem on the TI-89

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.

p

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