Leena loves peanut butter. today, she ate four tablespoons of it. if there are 6 mg of vitamin e in four tablespoons of peanut b
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To check for continuity at the edges of each piece, you need to consider the limit as approaches the edges. For example,
has two pieces, and
, both of which are continuous by themselves on the provided intervals. In order for
to be continuous everywhere, we need to have
By definition of , we have
, and the limits are
The limits match, so is continuous.
For the others: Each of the individual pieces of are continuous functions on their domains, so you just need to check the value of each piece at the edge of each subinterval.
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)=
∑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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