Diane Yeoman Definite Integral The Fundamental Theorem.

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Presentation transcript:

Diane Yeoman Definite Integral The Fundamental Theorem

Diane Yeoman Integrals and Area IIIIntegrals are used to find the area under a curve AAA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 Integral Since the integral has a definite start and stop point.

Diane Yeoman Definite Integral  A definite integral is shown as: a b f(x) dx a b f(x) dx Where a = the start point on the x-axis and b = the end 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=ax=b On this curve, f(x), we can find the area under the curve using a definite integral From x=a To x=b A definite integral is shown as: a b f(x) dx a b f(x) dx

Diane Yeoman Fundamental Theorem  A definite integral is evaluated as: a b f(x) dx = F(b) – F(a) a b f(x) dx = F(b) – F(a)Where F(b) is the integral of f(x) at the point x=b, and 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 F(a) is the integral of f(x) at the point x=a

Diane Yeoman Fundamental Theorem: Example  Calculate the definite integral: 0 5 e -2x dx 0 5 e -2x dx = (-1/2)e -2x | 0 5 Find F(5): =(-1/2) e -2(5) =(-1/2) e -10 =(-1/2) e -10 Find F(0): =(-1/2) e -2(0) =(-1/2) e 0 =(-1/2) e 0 =-1/2 =-1/2 Solve F(5) – F(0) =(-1/2) e (1/2)