Sketch the graph y = |x - 5|. Evaluate ∫|x - 5| dx x ∈ (1, 0). What does this value of the integral represent on the graph?

Question

Sketch the graph y = |x - 5|. Evaluate \(\int\limits_0^1|x-5|dx.\) What does this value of the integral represent on the graph?

Answer 1

Given equations are:

y = |x - 5|

y₁ = x - 5, if x - 5 ≥ 0

y₁ = x - 5 …… (1), if x ≥ 5

And y₂ = - (x - 5), if x - 5 < 0

y₂ = - (x - 5) …… (2), if x < 5

So, equation (1) is straight line that passes thorough (5, 0). Equation (2) is a line passing through (5, 0) and (0, 5). So, the graph of which is as follows:

\(\int^1_0 |x-5|dx\)

\(=\int^1_0 y_2dx\) (As when x is between (0,1) the given equation becomes y = - (x - 5) as shown in equation (2) shown ass shaded region in the above graph)

Now integrating by applying power rule, we get

Now applying the limits we get

Hence the value of \(\int^1_0 |x-5|dx\) represents the area of the shaded region OABC (as shown in the graph) and is equal to \(\frac{9}{2}\) square units.