Definite Integrals
DEFINITE means it's defined
, means both two boundaries are constant numbers.
Definite integrals properties
Definite integrals properties
Refer to Khan academy article: Definite integrals properties review
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Definite integral ←→ Limit of Riemann Sum
Definite integral ←→ Limit of Riemann Sum
Example
Solve:
It's easy to find
Δx = (π-0)/n = π/n
And
x𝖎 = S(𝖎) = a + 𝖎·Δx = 0+𝖎·Δx = 𝖎·π/n
So the result is:
Example
Solve:
Look at the boundaries, it's from
0 -> 5
,So the
Δx
must be cut ton
pieces, whereasΔx = (5-0)/n = 5/n
From the definition, We know the function
f(x) = x+1
To fill in the
x𝖎
inf(x𝖎)
, we need to figure out the sequence:Sequence
x𝖎 = S(𝖎) = a+𝖎·Δx
, and sincea
represents the bottom boundary,So
x𝖎 = 𝖎(0+Δx) = 𝖎·5/n
Get
x𝖎
back inf(x)
to have:
Example
We could easily get that the
Δx = 5/n
And the function is
f(x) = ln(x)
Since the
Δx
comes from Top & Bottom boundaries,So
Δx = (Top - Bottom)/n = 5/n = (Top - 2)/n
,And we get the
Top = 7
, and the Definite Integral is:
Example
See the
i=1
means it's usingRight Riemann Sum
, so the integral would be:The
Δx = 9/n
is easily seen.And we need to get the Sequence
x𝖎 = S(𝖎) = a + (𝖎-1)·Δx = (𝖎-1)·9/n
What we got there above, tells us
a=0
.According to that
Δx = (b-a)/n = (b-0)/n = 9/n
, we getb = 9
So the answer is:
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