Video: Finding an Interval given the Function's Average Value over It

Video Transcript

Convert the equation 𝑥 squared plus 𝑦 squared equals 25 into polar form.

Remember, we convert polar coordinates to Cartesian or rectangular coordinates using the formulae 𝑥 equals 𝑟 cos 𝜃 and 𝑦 equals 𝑟 sin 𝜃. And these are suitable for all values of 𝑟 and 𝜃. In our original equation, we’ve got 𝑥 squared and 𝑦 squared. So let’s use our formulae for 𝑥 and 𝑦 to generate expressions for 𝑥 squared and 𝑦 squared in terms of 𝑟 and 𝜃.

Since 𝑥 is equal to 𝑟 cos 𝜃, it follows that 𝑥 squared must be 𝑟 cos 𝜃 all squared, which we can distribute and say that 𝑥 squared is equal to 𝑟 squared times cos squared 𝜃. Similarly, we see that 𝑦 squared must be equal to 𝑟 sin 𝜃 all squared, which is equal to 𝑟 squared sin squared 𝜃.

Now our original equation says that the sum of these is equal to 25. So we can say that 𝑟 squared cos squared 𝜃 plus 𝑟 squared sin squared 𝜃 equals 25. Our next step is to factor 𝑟 squared on the left-hand side of this equation. So 𝑟 squared times cos squared 𝜃 plus sin squared 𝜃 equals 25. But why did we do this?

Well, here is where it’s useful to know some of our trigonometric identities by heart. We know that cos squared 𝜃 plus sin squared 𝜃 is equal to one for all values of 𝜃. So we can replace cos squared 𝜃 plus sin squared 𝜃 in our equation with one. So 𝑟 squared times one equals 25. Well, we don’t need this one. 𝑟 squared is simply equal to 25. We solve this equation by taking the square root of both sides. And we find that 𝑟 is equal to five.

Remember, we would usually take both the positive and negative square root of 25. But since 𝑟 represents a length, we don’t need to. 𝑥 squared plus 𝑦 squared equals 25 is the same as 𝑟 equals five in polar form.