The Definite Integral
6
Riemann Sums Consider any function f (x ) where a ≤ x ≤ b. Partition [a, b] into n subintervals of equal length Δx = b - a n Evaluate the function f (x ) at sample points c 1 , c 2 , ... c n chosen inside each subinterval. x 1 x 2 x 3 a = x 0 X Negative values Positive values c 1 c 2 c 3 c 4 c 5 c 6 x 4 x 5 b = x 6 Y Form the Riemann sum: f (c 1 ) · Δx + f (c 2 ) · Δx + ... + f (c n ) · Δx
Transcript of The Definite Integral
Riemann SumsConsider any function f (x) where a ≤ x ≤ b.
Partition [a, b] into n subintervals of equal length ∆x =b − a
nEvaluate the function f (x) at sample points c1, c2, ... cn
chosen inside each subinterval.
x1 x2 x3a = x0
X
Negative values
Positive values
c1 c2c3 c4 c5 c6
x4 x5 b = x6
Y
Form the Riemann sum:
f (c1) ·∆x + f (c2) ·∆x + ... + f (cn) ·∆x
Riemann Sum: [f (c1) + f (c2) + ... + f (cn)] ·∆x
x1 x2 x3a = x0
X
Negative values
Positive values
c1 c2c3
c4 c5 c6
x4 x5b = x6
O
Y
The Definite IntegralNotation:
f (c1) ·∆x + f (c2) ·∆x + ... + f (cn) ·∆x =n∑
i=1
f (ci ) ·∆x
Consider more and more subintervals. In other words, let thenumber n be larger and larger:
X
Y
Then the limit of the Riemann sums, as the number ofintervals becomes larger and larger is called The DefiniteIntegral or The Riemann Integral of f (x) from a to b:∫ b
af (x) dx = lim
n→∞
n∑i=1
f (ci ) ∆x
Geometric Interpretation of the Definite Integral
If f (x) ≥ 0 , which means that the graph of f (x) is above
the x - axis:
∫ b
af (x) dx = area under the graph and above the x axis
x
y
Geometric Interpretation of the Definite Integral
If f (x) is not positive (or negative) which means that thegraph of f (x) has parts both above and below the x axis:
∫ b
af (x) dx = area above - area below the x -axis
A2
A1
x
y
Geometric Interpretation of the Definite IntegralMore generally:
x
y
∫ b
af (x) dx = area above - area below the x -axis∫ b
af (x) dx = yellow area - blue area
