Circle A -- center(2, 0), radius 8 Circle A' -- center(-1, 5), radius 3
The expression of integral as a limit of Riemann sums of given integral 
is 4 
∑
from i=1 to i=n.
Given an integral .
We are required to express the integral as a limit of Riemann sums.
An integral basically assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinite data.
A Riemann sum is basically a certain kind of approximation of an integral by a finite sum.
Using Riemann sums, we have :
=∑f(a+iΔx)Δx ,here Δx=(b-a)/n
=f(x)=
⇒Δx=(5-1)/n=4/n
f(a+iΔx)=f(1+4i/n)
f(1+4i/n)=![[n^{2}(n+4i)]/2n^{3}+(n+4i)^{3}](https://tex.z-dn.net/?f=%5Bn%5E%7B2%7D%28n%2B4i%29%5D%2F2n%5E%7B3%7D%2B%28n%2B4i%29%5E%7B3%7D)
∑f(a+iΔx)Δx=
∑
=4∑
Hence the expression of integral as a limit of Riemann sums of given integral is 4 ∑ from i=1 to i=n.
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The present value of the loan is R576923.
<h3>What is compound interest ?</h3>
Compound interest is giving the current instalment in terms of the total previous amount.
The formula is given by
A = P(1 + r/100)ⁿ.
Where,
A = Amount, P = Principle, r = rate of interest, n = Time in years.
In case the compound if interest is given every two months that is 6 instalments each year the above given formula will be
A = P{ 1 + (r/6)/100 }⁶ⁿ.
According to the given question
Rate(r) = 7.5%
Time(n) = 6 years
The loan will be paid back in 6 years every second month and it is compounded.
∴ No. of instalments = (12 × 6)/2
= 36.
Now each instalments is of R25000
So, The total amount she has to pay back to his father is
= (25000 × 36)
= R900000.
We know compounding every two months is
A = P(1 + r/100)ⁿ
900000 = P{ 1 + (7.6/6)/100 }³⁶
900000 = P{ 1 + 1.25/100 } ³⁶
900000 = P(1.0125)³⁶
900000 = 1.56P
∴ P = 576923.
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1/10 of all take 2/5 time
10/10=10 times 1/10
so 10 times 2/5 time=20/5=4 days to finish 1 order
4 days
