`int_0^10|x - 5|dx` Evaluate the integral by interpreting it in terms of areas.

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Textbook Question

Chapter 5, 5.2 - Problem 40 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lemjay | High School Teacher | (Level 3) Senior Educator

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`int_0^10 |x-5|dx`

To interpret this in terms of area, graph the integrand. The integrand is the function f(x) = |x - 5|.

Then, shade the region bounded by f(x) = |x-5| and the x-axis in the interval [0,10]. (Refer to the attached figure.)

Notice that the bounded region forms two right triangles. So to evaluate the integral, compute the area of each triangle and take the sum.

`int_0^10 |x-5| dx`

`=A_(Delta_1)+ A_(Delta_2)`

`=1/2b_1h_1+1/2b_2h_2`

`=1/2*5*5 + 1/2*5*5`

`=25/2+25/2`

`=50/2`

`=25`

Thus,  `int_0^10 |x-5| dx = 25` .

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