I have written up the solution on the whiteboard in the video attached and explained the steps.
Vina M.
asked 01/23/25Integration question
I keep getting wrong answer, can someone help me with tis
2 Answers By Expert Tutors
Ali I. answered 01/23/25
Master's in Mathematics with 5+ years Tutoring Experience
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 + ∫51/2 -(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.
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Vina M.
Thank you.01/23/25