Area and the Definite Integral Michael Freeze MAT 151 UNC - - PowerPoint PPT Presentation

Area and the Definite Integral

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

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Section 7.3 :: Area and the Definite Integral

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Archimedes’ Quadrature of the Parabola

• Archimedes was able to show that the area of a

parabolic section is 4

3 times the area of an

inscribed triangle.

• A key component of his method was the

summation of a geometric series.

• The method did not extend to the quadrature
• f the circle.

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The Definite Integral

We write b

a f (x) dx to denote the signed area

between the graph of y = f (x) and the x-axis over the interval a ≤ x ≤ b. The value of the definite integral is given by b

a

f (x) dx = lim

n→∞

n

• i=1

f (xi) ∆x

• ,

provided the limit exists, where ∆x = b−a

n

and xi is any value of x in the ith subinterval of [a, b].

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Total Change in F(x)

If f (x) gives the rate of change of F(x) for x in [a, b], then the total change in F(x) as x goes from a to b is given by lim

n→∞

n

• i=1

f (xi) ∆x

• =

b

a

f (x) dx.

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Oil Leakage

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