Leibniz (1646-1716) first discovered calculus

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• Leibniz (1646-1716) first discovered calculus
• the key idea was that differentiation and
  integration undo each other
• (The Fundamental Theorem of Calculus)

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• Newton (1642-1727) first discovered calculus
• the key idea was that differentiation and
  integration undo each other
• (The Fundamental Theorem of Calculus)

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• Leibniz invented the notation of
• d and

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• Integration
• is needed to find the area under a curve

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R
To find the area of region R, two definite
integrals have to be calculated
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• The Trapezium Rule always gives an approximation
  less than the true value

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• The area between the curve f(x), the
• x-axis, x 1 and x 4 is given by

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• The area below the x-axis is always negative

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• The Trapezium Rule using 10 strips will give a
  more accurate approximation than one using 5
  strips

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• Cavalieri was the first to think of an area as
  being made up of the sum of an infinite number of
  components
• He showed, using these methods, that the integral
  of xn from 0 to a was an1/(n 1)

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NEVER
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TRUE
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SOMETIMES
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