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Length of Curve by Integrals

alane1994

Active member
Oct 16, 2012
126
I had a question on a quiz that I missed... I am unsure how they got this answer. If someone could explain it would be great!

Write the integral that gives the length of the curve.

[tex]y=f(x)=\int_{0}^{4.5x} \sin{t} dt[/tex]

It was multiple-choice(multiple-guess;)).

[tex] \text{Choice A } L=\int_{0}^{\pi} \sqrt{1+4.5(\sin(x))^2}dx[/tex]

[tex] \text{Choice B } L=\int_{0}^{\pi} \sqrt{1+20.25(\sin(4.5x))^2}dx[/tex]

[tex] \text{Choice C } L=\int_{0}^{\pi} \sqrt{1+20.25(\sin(x))^2}dx[/tex]

[tex] \text{Choice D } L=\int_{0}^{\pi} \sqrt{1+4.5(\sin(4.5x))^2}dx[/tex]

The correct answer is B... any way to explain in everyday people speak?
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
I had a question on a quiz that I missed... I am unsure how they got this answer. If someone could explain it would be great!

Write the integral that gives the length of the curve.

[tex]y=f(x)=\int_{0}^{4.5x} \sin{t} dt[/tex]

It was multiple-choice(multiple-guess;)).

[tex] \text{Choice A } L=\int_{0}^{\pi} \sqrt{1+4.5(\sin(x))^2}dx[/tex]

[tex] \text{Choice B } L=\int_{0}^{\pi} \sqrt{1+20.25(\sin(4.5x))^2}dx[/tex]

[tex] \text{Choice C } L=\int_{0}^{\pi} \sqrt{1+20.25(\sin(x))^2}dx[/tex]

[tex] \text{Choice D } L=\int_{0}^{\pi} \sqrt{1+4.5(\sin(4.5x))^2}dx[/tex]

The correct answer is B... any way to explain in everyday people speak?
Hi alane1994, :)

The arc length of the graph of the function \(f\) between the points \(x=a\) and \(x=b\) is given by,

\[s = \int_{a}^{b} \sqrt { 1 + [f'(x)]^2 }\, dx\]

Refer: Arc length - Wikipedia, the free encyclopedia

So first you'll have to find \(f'(x)\). Can you give it a try? :)

Kind Regards,
Sudharaka.