How do you evaluate the definite integral #int abs(x-5) dx# from #[0,10]#?

Answer 1

25

#int_0^10 abs(x-5) \ dx#
#abs(x-5) = x-5, x in [5,oo)#
#abs(x-5) = -(x-5), x in [5,-oo)#
#implies int_0^5 -(x-5) \ dx + int_5^10 x-5 \ dx#
#= [ (x-5)^2/2 ]_5^0 + [ (x-5)^2/2 ]_5^10#
#= (25/2 ) + ( 25/2 ) = 25#

you can see this in the graph

graph{abs(x-5) [-5.375, 14.625, -2.16, 7.84]}

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Answer 2

To evaluate the definite integral (\int_{0}^{10} |x - 5| , dx), we first need to identify the intervals where the function inside the absolute value changes its behavior. In this case, (|x - 5|) changes from (x - 5) to (-(x - 5)) at (x = 5).

So, we split the integral into two parts:

  1. From (x = 0) to (x = 5): (\int_{0}^{5} (5 - x) , dx)
  2. From (x = 5) to (x = 10): (\int_{5}^{10} (x - 5) , dx)

Integrating each part:

  1. (\int_{0}^{5} (5 - x) , dx = [5x - \frac{x^2}{2}]_{0}^{5} = (5(5) - \frac{5^2}{2}) - (5(0) - \frac{0^2}{2}) = (25 - \frac{25}{2}) - (0) = \frac{25}{2})
  2. (\int_{5}^{10} (x - 5) , dx = [\frac{x^2}{2} - 5x]_{5}^{10} = (\frac{10^2}{2} - 5(10)) - (\frac{5^2}{2} - 5(5)) = (50 - 50) - (\frac{25}{2} - 25) = 0 - (\frac{25}{2} - 25) = -(\frac{25}{2} - 25))

Adding both parts:

(\frac{25}{2} - (\frac{25}{2} - 25) = \frac{25}{2} - \frac{25}{2} + 25 = 25)

Therefore, the value of the definite integral (\int_{0}^{10} |x - 5| , dx) is (25).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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