diane yeoman definite integral the fundamental theorem

Diane Yeoman

Definite IntegralThe Fundamental Theorem

Diane Yeoman

Integrals and Area

Integrals are used to find the area under a curve

As we used with our Reimann sum, we need to know where on the x-axis to start and stop our search for the area.When we do this, it is called a

Definite IntegralSince the integral has a definite start and

stop point.

Diane Yeoman

Definite Integral

A definite integral is shown as:

ab f(x) dx

Where a = the start point on the x-axis and b = the end point on the x-axis

This will replace the Riemann Sum method of finding the area under a curve.

Diane Yeoman

Definite Integral: Area under a curve

x=a x=b

On this curve, f(x), we can find the area under the curve using a definite integralFrom x=aTo x=bA definite integral is shown as:

ab f(x) dx

Diane Yeoman

Fundamental Theorem

A definite integral is evaluated as:

ab f(x) dx = F(b) – F(a)

Where F(b) is the integral of f(x) at the point

x=b, and F(a) is the integral of f(x) at the point

x=a

Diane Yeoman

Fundamental Theorem: Example

Calculate the definite integral:

05 e-2x dx

= (-1/2)e-2x |0

5

Find F(5): =(-1/2) e-2(5) =(-1/2) e-10

Find F(0): =(-1/2) e-2(0)

=(-1/2) e0

=-1/2Solve F(5) – F(0)

=(-1/2) e-10 - (1/2)