For this type of question, do you need a graph to figure out the solution or is there another non graphical way to know which is the top and bottom function? I used desmos to graph the functions but how would you do this question without looking at desmos?

Step-by-step explanation:

First, find the intersections.

4x = x³

0 = x³ − 4x

0 = x (x² − 4)

0 = x (x + 2) (x − 2)

x = -2, 0, 2

So we will consider two intervals: -2 ≤ x ≤ 0, and 0 ≤ x ≤ 2.

Pick a point within each interval and evaluate both functions at that point.  Compare to see which function is greater.

At x = -1, y₁ = 4x = -4, and y₂ = x³ = -1.  So y₂ > y₁.

At x = 1, y₁ = 4x = 4, and y₂ = x³ = 1.  So y₁ > y₂.

So the integral would be:

∫₋₂⁰ (x³ − 4x) dx + ∫₀² (4x − x³) dx

the answer is 62.8 cubic units