![Riemann Sums
Consider any function f (x) where a ≤ x ≤ b.
b−a
Partition [a, b] into n subintervals of equal length ∆x =
n
Evaluate the function f (x) at sample points c1 , c2 , ... cn
chosen inside each subinterval.
Y
Positive values
a x0 x1 x2 x3 x4 x5 b x6
c1 c2 c3 c4 c5 c6
X
Negative values
Form the Riemann sum:
f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x](https://image.slidesharecdn.com/definite-integral-100519115058-phpapp01/85/The-Definite-Integral-1-320.jpg)
![Riemann Sum: [f (c1 ) + f (c2 ) + ... + f (cn )] · ∆x
Y
Positive values
a x0 x1 x2 x3 x4 x5 b x6
c1 c3 c4 c5 c6
O c2 X
Negative values](https://image.slidesharecdn.com/definite-integral-100519115058-phpapp01/85/The-Definite-Integral-2-320.jpg)
The Definite Integral
- 1. Riemann Sums Consider any function f (x) where a ≤ x ≤ b. b−a Partition [a, b] into n subintervals of equal length ∆x = n Evaluate the function f (x) at sample points c1 , c2 , ... cn chosen inside each subinterval. Y Positive values a x0 x1 x2 x3 x4 x5 b x6 c1 c2 c3 c4 c5 c6 X Negative values Form the Riemann sum: f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x
- 2. Riemann Sum: [f (c1 ) + f (c2 ) + ... + f (cn )] · ∆x Y Positive values a x0 x1 x2 x3 x4 x5 b x6 c1 c3 c4 c5 c6 O c2 X Negative values
- 3. The Definite Integral Notation: n f (c1 ) · ∆x + f (c2 ) · ∆x + ... + f (cn ) · ∆x = f (ci ) · ∆x i=1 Consider more and more subintervals. In other words, let the number n be larger and larger: Y X Then the limit of the Riemann sums, as the number of intervals becomes larger and larger is called The Definite Integral or The Riemann Integral of f (x) from a to b: b n f (x) dx = lim f (ci ) ∆x a n→∞ i=1
- 4. Geometric Interpretation of the Definite Integral If f (x) ≥ 0 , which means that the graph of f (x) is above the x - axis: b f (x) dx = area under the graph and above the x axis a y x
- 5. Geometric Interpretation of the Definite Integral If f (x) is not positive (or negative) which means that the graph of f (x) has parts both above and below the x axis: b f (x) dx = area above - area below the x -axis a y A1 x A2
- 6. Geometric Interpretation of the Definite Integral More generally: y x b f (x) dx = area above - area below the x -axis a b f (x) dx = yellow area - blue area a