Question

Sketch the graph y = | x − 5 |. Evaluate \[\int\limits_0^1 \left| x - 5 \right| dx\]. What does this value of the integral represent on the graph.

Answer 1

We have,
y = | x − 5 | intersect x = 0 and x = 1 at (0, 5) and (1, 4)
Now,
\[y = \left| x - 5 \right|\]
\[ = - \left( x - 5 \right)\text{ For all }x \in \left( 0, 1 \right)\]
Integration represents the area enclosed by the graph from x = 0 to x = 1
\[A = \int_0^1 \left| y \right| d x\]
\[ = \int_0^1 \left| x - 5 \right| d x\]
\[ = \int_0^1 - \left( x - 5 \right) d x\]
\[ = - \int_0^1 \left( x - 5 \right) d x\]
\[ = - \left[ \frac{x^2}{2} - 5x \right]_0^1 \]
\[ = - \left[ \left( \frac{1}{2} - 5 \right) - \left( 0 - 0 \right) \right]\]
\[ = - \left( - \frac{9}{2} \right)\]
\[ = \frac{9}{2}\text{ sq . units }\]