Circumference of a circle Use calculus to find the...

Key Concepts

Definite Integration

Definite integration is a fundamental concept in calculus that involves integrating a function over a specific interval to accumulate a total quantity. In the context of finding arc lengths, definite integrals are used to sum the lengths of infinitesimal segments to obtain the total length of a curve.

Arc Length

Arc length refers to the concept of finding the length of a curve by integrating infinitesimally small segments along the curve. In calculus, this typically involves computing an integral of the form ds = ?((dx/dt)² + (dy/dt)²) dt over the parameter interval, which accumulates the total length of the path.

Parametric Equations

Parametric equations are used to represent curves by expressing the coordinates in terms of a parameter, such as t. This approach is particularly useful for curves that are hard to describe using a single function and facilitates the application of calculus techniques, like differentiation and integration, to determine geometric properties.

Transcript

00:01 In this problem, we're asked to find the circumference of a circle with radius a as pictured on the right, using calculus.

00:08 So firstly, let's think about what circumference is.

00:12 You'll realize that it's actually the same as arc length, so what we can do is use integration to find the arc length of a circle with radius a.

00:19 So the first thing is to parameterize it in terms of t.

00:23 Well, you can think back to a unit circle, which is a circle with radius 1.

00:29 We have x equals cosine of theta, y equals sine of theta.

00:35 Therefore, if you think about it, the radius a, when the radius is a, the maximum value for sign, let's say, when if the radius is 3, will be 3.

00:47 Therefore, in order to make, at this point, we're at pi over 2, in order to make sign of pi over 2 equal to 3, you would have to multiply it by the radius itself.

00:59 Therefore, you can say y equals a sine theta and x equals acosta theta, where a is the radius.

01:08 Great.

01:09 So now we're able to parameterize our circle in terms of t.

01:11 So step one, x equals a cosine of t, and then y can equal a sign of t.

01:21 And it's also important to write the bounds for t.

01:24 So we're going to say t is in between zero and two pi, like the unit, is.

01:32 Now we have to calculate ds using the formula.

01:35 You'll realize that we first, however, need to calculate dx over d t and d y over d t, which are pretty doable.

01:42 So the derivative of x with respect to t will be negative a sign of t.

01:52 And for y with respect to t, it will be simply a cosine of t.

01:59 Now we're able to easily.

02:01 Calculate ds using the formula that i wrote on the right...

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