What is the formula for finding arclength when given an integral?

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In summary, to find the arclength of the given curve, y = integral from -pi/2 to x of sqrt(cost)dt, we can use the fundamental theorem of calculus. This involves taking the derivative of the integral, which simplifies to sqrt(cosx). The constant -pi/2 does not affect the final result.
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JJ6
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Homework Statement



Find the arclength of the curve given by y= integral from -pi/2 to x of sqrt(cost)dt. X is restricted between -pi/2 and pi/2.

Homework Equations



L = Integral from a to b of sqrt((dy/dx)^2 + 1)dx
L = Integral from a to b of sqrt((dy/dt)^2 + (dx/dt)^2)dt

The Attempt at a Solution



I'm not even sure how to start this problem since integral of sqrt(cost) has no simple function representing it. Can somebody please give me some direction?
 
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  • #2
It is an application of the fondamental theorem of calculus.

[tex]\frac{d}{dx}\int_{x_0}^xf(t)dt=f(x)[/tex]
 
  • #3
So does that mean that d/dx of the integral from -pi/2 to x of sqrt(cost)dt = sqrt(cosx)?

Does the -pi/2 just disappear because it is a constant?
 
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  • #4
JJ6 said:
So does that mean that d/dx of the integral from -pi/2 to x of sqrt(cost)dt = sqrt(cosx)?

Yep.

JJ6 said:
Does the -pi/2 just disappear because it is a constant?
Disapear from where?
 

What is a Complicated Arclength Problem?

A Complicated Arclength Problem is a mathematical problem that involves calculating the length of a curve or arc on a complicated shape, such as a spiral or irregularly-shaped curve. It requires the use of advanced mathematical techniques such as calculus and geometry.

Why are Complicated Arclength Problems important?

Complicated Arclength Problems have many real-world applications, such as in engineering and physics. They allow us to accurately measure the length of complex shapes and curves, which is crucial in designing and building structures and machines.

What are some common techniques used to solve Complicated Arclength Problems?

Some common techniques used to solve Complicated Arclength Problems include integration, parametric equations, and the Pythagorean theorem. These techniques allow us to break down the complex shape into smaller, more manageable parts and use mathematical formulas to find the length of each part.

Can Complicated Arclength Problems be solved without using advanced mathematics?

No, Complicated Arclength Problems cannot be solved without the use of advanced mathematics. These problems require the use of calculus and geometry, which are branches of mathematics that deal with complex shapes and curves.

What are some tips for solving Complicated Arclength Problems?

Some tips for solving Complicated Arclength Problems include breaking down the shape into smaller, simpler parts, using the correct formulas and equations, and checking your work for accuracy. It is also important to have a strong understanding of calculus and geometry concepts.

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