Video: Evaluating the Definite Integral of a Function Containing Root Using Integration by Substitution

Convert the parametric equations π‘₯ = 𝑑² + 2 and 𝑦 = 3𝑑 βˆ’ 1 to rectangular form.

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Video Transcript

Convert the parametric equations π‘₯ equals 𝑑 squared plus two and 𝑦 equals three 𝑑 minus one to rectangular form.

Here, we have a pair of parametric equations. We have π‘₯ is equal to some function of 𝑑 and 𝑦 is equal to some other function of 𝑑. To convert parametric equations to rectangular form, we need to find a way to eliminate the 𝑑. So looking at our equations, we can see that we can rearrange the equation in 𝑦 to make 𝑑 the subject. We begin by adding one to both sides. And then, we divide through by three. So we see that 𝑑 is equal to 𝑦 plus one all over three.

Now, we go back to our equation for π‘₯. We replace 𝑑 with 𝑦 plus one over three. And we find that π‘₯ equals 𝑦 plus one over three all squared plus two. And there will be certain circumstances where we’re required to distribute the parentheses and simplify. In this case, that’s not necessary. And so, we’re finished. We’ve converted the parametric equations π‘₯ equals 𝑑 squared plus two and 𝑦 equals three 𝑑 minus one into rectangular or Cartesian form. It’s π‘₯ equals 𝑦 plus one over three all squared plus two.

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