Evaluate `int _(-2)^8 |x| dx` Evaluate ʃ (superscript 8)(subscript -2) abs(x) dx and sket a corresponding graph to ensure that answer is reasonable.

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

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`|x|=x` if `x>=0`
`|x|=-x` if `x<=0`

Thus, if we split the integral into two pieces, we will be able to rewrite `|x|` as something more useful:

`int _(-2)^8 |x| dx = int _(-2)^0 |x| dx + int _(0)^8 |x| dx`

`= int _(-2)^0 -x dx + int _(0)^8 x dx`

`= -(1/2)x^2 |_(-2)^0 + (1/2)x^2 |(0)^8`

`=[-(1/2)0^2]-[-(1/2)(-2)^2]+[(1/2)8^2]-[(1/2)0^2]=0+(1/2)(4)+(1/2)(64)-0=34`

To see if this is reasonable, consider the graph:

 

We can actually just calculate the area under the curve using the formula for the area of a triangle:

The small triangle (on the left) has height 2, base 2, so its area is: `(1/2)2*2=2`

The large triangle (on the right) has height 8, base 8, so its area is: `(1/2)8*8=32`

So the total area is 2+32=34, which is what we had from the integral.

 

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