Q 42. - Calculus

Question

In Exercises 32–47 convert the equations given in polar coordinates to rectangular coordinates.

$r^2=\sin \theta$

Step-by-Step Solution

Answer

The required equation is ${\left(x^2+y^2\right)}^{\frac{3}{2}}=y$.

Step 1. Given information.

The given equation in polar coordinates is:

$r^2=\sin \theta$

Step 2. Find the equation in rectangular coordinates.

$r^2=\sin \theta$

First, multiply both sides by r,

$$\begin{gathered} r·r^2=r \sin \theta \\ {r}^3=y \left[r \sin \theta =y\right] \\ {\left(\sqrt{x^2+y^2}\right)}^3=y \left[r^2=x^2+y^2 \mathrm{and} r=\sqrt{x^2+y^2}\right] \\ {\left(x^2+y^2\right)}^{\frac{3}{2}}=y \end{gathered}$$

Therefore, the equation in rectangular coordinates is ${\left(x^2+y^2\right)}^{\frac{3}{2}}=y$.