If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Calculus 2

### Unit 5: Lesson 9

Arc length: polar curves

# Arc length of polar curves

Sal shows the polar arc length formula, and explains why it is true.

## Want to join the conversation?

• Why is x= rcos(theta) and why is y=rsin(theta)?
• It is derived from the trig functions in a polar graph.
cos Θ = adjacent/hypotenuse = x/r
sin Θ = opposite/hypotenuse = y/r
• Why are we converting Polar to Cartesian form ( ) ? Y can't we just use ds (arc length) = rdθ and integrate it?
• Since `r=f(θ)`, the length of the arc is not given by `r*θ`, (that is only valid on segments of circles).
• I think the question people are asking, which has still not been properly answered, relates to the difference between the Area derivation and the Arc Length derivation. In both, r = f(t), so radius can change over the function of the angle, but when integrating to find the Area this doesn't matter, but for the Arc length it does. Why?
• Why doesn't using the circumference of a infinitesimally small sector of a circle and summing the circles up work?
integral[a,b][(2*pi*r *d(theta)/(2*pi)]=integral [a,b][r*d(theta)]?
• since s=rΘ why isnt or why can't arc length be ds=rdΘ in the same way for how area enclosed in polar functions is dA=(r^2)/2 dΘ
• Because the radius changes over a function of the angle, hence r= f(Θ). When the radius is constant, like a circle fixed at the origin for example, the formula reduces down to the S=Θr.
• I was wondering how the calculus equation for arc length relates to the algebra version: s=r(theta) ? I have attempted to find a connection but cannot seem to find one.
• In case of common equation, not polar curves, can we also figure out the length of line between two points using the integral calculus?
ex) y=f(x) x=[a, b]
(1 vote)
• This polar graph stuff seems to have popped up out of nowhere for me along this integration playlist. Is there like an introduction to polar graphs or the like somewhere in KA?
(1 vote)
• Why can't I integrate "arc length = f(theta)* (d theta) " to find arc length?
I think it's a good approximation that arc length = f(theta)* (d theta)
Also, when we calculate the area of the polar graph, we use "(1/2)(f(theta)^2)(d theta)" to approximate the area of the curve. I think this two are similar, but why arc length can't be found by similar method but area can
(1 vote)
• arc length = Integral( r *d(theta)) is valid only when r is a constant over the limits of integration, as you can test by reducing the general formula from this video when dr/d(theta) =0. In general r can change with theta. In Sal's video he could have constructed a different right angled triangle with ds as the hypotenuse and the other two sides of lengths dr and r*d(theta). I will leave the construction of this triangle as an intellectual exercise :-) Hint: use polar coordinates.
(1 vote)
• In the process of adding dx^2 and dy^2 you have two dtheta^2 to add together. It makes sense that the result of this addition should be 2dtheta^2. Wouldn't it be like adding x^2 and x^2? The result of that addition is 2x^2. So why wouldn't dtheta^2 plus dtheta^2 equal 2dtheta^2?
(1 vote)
• What you say is correct. dtheta^2 is just a variable that can be added like any other.
In the present context of the video though, sal never added two dtheta^2 's together.
Notice that the dtheta^2 was factored out and had the other things as its coefficients.
(1 vote)