How do you evaluate the integral of absolute value of (x - 5) from 0 to 10 by finding area?

Question

Luke Alvarez · Accepted Answer

The region under the graph of #f(x)=|x-5|#  from #a=0# to #b=10# is made up of two triangles.  The triangles both have a base of length 5 units and a height of 5 units, so they each have an area of #\frac{1}{2}\cdot 5\cdot 5=\frac{25}{2}#.  Altogether the total area is 25, and this is the value of the definite integral #\int_{0}^{10}f(x)\ dx#. graph{|x-5| [-5.33, 14.67, -2.8, 7.2]}

Charles Arocha · Answer

undefined To evaluate the integral of the absolute value of $ |x - 5| $ from 0 to 10 by finding area, we split the integral into two parts: from 0 to 5 and from 5 to 10. Then, we integrate the function $ |x - 5| $ within each interval and sum up the areas. Since $ |x - 5| $ equals $ x - 5 $ for $ x \geq 5 $ and $ -(x - 5) $ for $ x < 5 $, we integrate $ x - 5 $ from 0 to 5 and $-(x - 5)$ from 5 to 10. The area is then the sum of these two integrals. So, the integral evaluates to $ \int_{0}^{5} (x - 5) \, dx + \int_{5}^{10} -(x - 5) \, dx $. Solving each integral separately, we get $ \frac{1}{2}(x^2 - 10x) $ from 0 to 5, and $ -\frac{1}{2}(x^2 - 10x) $ from 5 to 10. Plugging in the limits and subtracting, we find the total area.