Finding the Intersection of Two Graphs: Exact Solutions and Integrals

[Jbreezy]

Homework Statement

Hi I'm trying to find where these two graphs intersect I would like it to be exact but it isn't quite working.
If I have y = tan(x) and y = x^1/3 how can I solve exactly?

Homework Equations

tan(x) = x^1/3 ? Hmm.
I'm not sure. I don't want arctan popping up on the right side. So I don't know really.

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Hi I'm trying to find where these two graphs intersect I would like it to be exact but it isn't quite working.
If I have y = tan(x) and y = x^1/3 how can I solve exactly?

Homework Equations

tan(x) = x^1/3 ? Hmm.
I'm not sure. I don't want arctan popping up on the right side. So I don't know really.

The Attempt at a Solution

Show your work.

 

[Jbreezy]

I'm supposed to just set up the integral not evaluate it. They want me to consider y = tanx and y = x^1/3 in the first quadrant.
Rotated about x I have:
V = 2PI (integral) x(tanx-x^1/3) dx
Between what ever result I get for the intersection of 0 and tanx = x^1/3
and for about y I have
V = 2PI( integral) y(arctan(y) - y^3) dy
Between 0 and whatever y^3 = arctan(y) intersections is.

 

[Jbreezy]

This is my work
tan(x) = x^1/3
x = arctan(x^1/3) ok ... now I'm stuck

 

[Ray Vickson]

This is my work
tan(x) = x^1/3
x = arctan(x^1/3) ok ... now I'm stuck

That's because there is (very probably) no closed-form solution; just use a numerical method.

 

[Jbreezy]

What do you mean a numerical method and how am I supposed to indicate the upper bound my teacher gives 0's for decimals. Also does my integrals look OK?

 

[Jbreezy]

Do my integrals look set up properly? Also how am I supposed to write an upper limit? Like integral from 0 to
tan(x) = x^1/3? Because I can't really solve this but I have to represent it exactly.