Definite Integrals
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Last updated
DEFINITE means it's defined, means both two boundaries are constant numbers.
Definite integrals propertiesRefer to Khan academy article: Definite integrals properties review
Definite integral ←→ Limit of Riemann SumSolve:
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:
Solve:
Look at the boundaries, it's from 0 -> 5,
So the Δx must be cut to n pieces, whereas Δx = (5-0)/n = 5/n
From the definition, We know the function f(x) = x+1
To fill in the x𝖎 in f(x𝖎), we need to figure out the sequence:
Sequence x𝖎 = S(𝖎) = a+𝖎·Δx, and since a represents the bottom boundary,
So x𝖎 = 𝖎(0+Δx) = 𝖎·5/n
Get x𝖎 back in f(x) to have:
Solve:
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:
Solve:
See the i=1 means it's using Right 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 thatΔx = (b-a)/n = (b-0)/n = 9/n, we get b = 9
So the answer is:
