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Evaluate the integral by interpreting it in terms of areas. 10 |x − 5| dx 0

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The integral is ∫(10 |x − 5| dx) from 0 to 10.

This expression can be interpreted in terms of areas as the area between the function y = 10 |x − 5| and the x-axis from x = 0 to x = 10.

Notice that the graph of |x - 5| is a V-shaped graph with its vertex at (5, 0), so the graph is symmetric about the line x = 5. Therefore, we can split the integral into two parts, from 0 to 5 and from 5 to 10.

When x is between 0 and 5, |x - 5| = 5 - x, so the integral becomes:

∫(10(5 - x) dx) from 0 to 5

= [10(5x - (x^2)/2)] from 0 to 5

= (125 - 125/2) - 0

= 62.5

When x is between 5 and 10, |x - 5| = x - 5, so the integral becomes:

∫(10(x - 5) dx) from 5 to 10

= [10((x^2)/2 - 5x)] from 5 to 10

= 0 - (125 - 125/2)

= -62.5

Therefore, the area between the function and the x-axis from x = 0 to x = 10 is:

62.5 + (-62.5) = 0

So, ∫(10 |x − 5| dx) from 0 to 10 = 0.

User Rob King
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