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Numerade Educator

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Problem 44 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int^{2}_{0} \mid 2x - 1 \mid \,dx $

Answer

$\frac{5}{2}$

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Video Transcript

this problem makes the most sense to Look at the graph of absolute value of two X -1. Yeah. Dx. And the reason for that is uh the graph goes below the X axis. If you were just a graph uh the curve two x minus one, it was down one and then go up to write one. Uh so like if you were to plug him one for X, you'd be at the order pair 11. But if you plugged in zero for X you will get negative one. But the absolute value of negative one will give you positive one. So the graph actually looks something like this. And uh if you plugged in to for ex you would I think it's pretty clear you would get three if you plug that in. Um So it makes the most sense defying the two separate areas. The area of this red triangle Which the area of triangles one half times the bait. The basis of width of 1/2 and the height is one. And then you want to add to it. I'll make this blue. Oh This blue triangle which again the area of China is one half the length of this base now is 3/2ves Times the height of three which we've already evaluated. So as you're looking at all this, you you actually figure do the math, you get 1/4 plus nine. Force, you know, multiplying fractions is easy. Just enumerated the numerator with numerator over denominator times denominator. And then you can add your numerator together because the denominators are the same. And then that reduces to 5/2 when you divide top and bottom by two. This is the correct answer. So let's move on.