Integration question

Question

I keep getting wrong answer, can someone help me with tis

Answer 1

I have written up the solution on the whiteboard in the video attached and explained the steps.

Answer 2

One way to understand absolute value is that it changes your function to become piecewise. An example of this is if I consider f(x)=|x|, then this can be considered as the

f(x)=x when x is positive and

f(x)=-x when x is negative (double negative forces it back to positive).

In this question you are considering |x-2x^2|. This is normally a parabola which faces downward, but due to the absolute value, the part of the parabola that is negative will be positive instead. This parabola has a vertex at (1/4, 1/8) and zeros at x=0, x=1/2.

This means that the function is negative from (-∞, 0), positive from (0, 1/2), and then negative from (1/2,∞).

So whenever we are on a negative interval, we should consider the opposite of the function:

-(x-2x^2) on (-∞, 0)

x-2x^2 on (0,1/2)

-(x-2x^2) on (1/2, ∞)

Now we can break up the integral based off this information

∫⁵_-1 |x-2x^2| dx= ∫⁰_-1 -(x-2x^2) dx+ ∫^1/2₀ x-2x^2 dx + ∫⁵_1/2 -(x-2x^2) dx

You then would evaluate each of these integrals separately. Let me know if you need help on that step, and sorry for the weird formatting.