The length of the curve from x = 3 to x = 6 is 192 units
The curve is given as:
from x = 3 to x = 6
Start by differentiating the curve function
Evaluate
The length of the curve is calculated using:
This gives
Expand
This gives
Express as a perfect square
Evaluate the exponent
Differentiate
Expand
L = (6³ + 6) - (3³ + 3)
Evaluate
L = 192
Hence, the length of the curve is 192 units
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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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