Question

What is the value of this integral `∫_0 ^10 |x−5|dx` ?

Answer 1

The value of the above integral is 25. It can be solved as follows:

Explanation:

Since we are integrating a modulus function here, we first look for its breaking point, that is the value of x, before and after which the value of |x-5| changes. Also one should keep in mind that modulus always gives positive values, while solving such questions.

• We know that (x-5) <0 for x<5.
  `implies` for x `in` [0,5] ; |x-5| = -(x-5)
• Also, for X>= 5, (x-5) >0.
  `implies` for x ranging from [5,10]; |x-5| = x-5 only as it is positive in this interval.

Now we break our integral into two parts:
`int_0^10|x-5|dx=int_0^5(5-x)dx+int_5^10(x-5)dx=|_0^5 (5x - x^2/2( + |_5^10(x^2/2-5x) = [25-25/2] + [(100/2-50) - (25/2 - 25)] = 25`