Is the domain of $y(x) = (5^{2/3} - x^{2/3})^{3/2}$ $[-5, 5]$ and if so why doesn't Maple plot the entire domain?

Question

On the platform MIT OCW, there is a course 18.01SC, single-variable calculus. There is a problem set with the following problem:

Find the area of the astroid $x^{2/3} +y^{2/3}=a^{2/3}$ revolved around the x-axis.

To solve this, we form an integral of arc length revolved around the x-axis: $$\int 2 \pi y \sqrt{1 + (y')^2} dx$$ The issue is in figuring out the interval of the integral.

If I solve the equation for $y$ using $a=5$ and ask Maple to plot it, I get:

Note that $$y(x) = (5^{2/3} - x^{2/3})^{3/2}$$

so we see that indeed for values of x larger than $|5|$ the value inside the root becomes negative so we are not in real numbers anymore. It makes sense to me that the domain of the function is $[-5,5]$.

Why doesn't Maple show values of the function for $x \in [-5,0)$?

Answer 1

You could read the Maple Help-pages for topics root (especially 4th and 5th bullet points), and surd.

plot((5^(2/3)-surd(x,3)^2)^(3/2), x=-5...5);

An alternative is to try and run your example under the RealDomain package. Note this difference in behavior:

restart;

(-8.0) ^ (2/3);

    -2.000000001 + 3.464101615 I

with(RealDomain):

(-8.0) ^ (2/3);

          4.000000000

After loading that package (only needed once per session) you could obtain the same plot as above, ie.

restart;
with(RealDomain):
plot((5^(2/3)-x^(2/3))^(3/2), x=-5...5);

This happens to handle your given example, but can have issues (that are more difficult to figure out, let alone resolve) with more involved examples. It's not my preferred approach.