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

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).

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step-by-step explanation: