Question

Evaluate the integral by interpreting it in terms of areas.

\(\int_0^{10} | x - 5|dx\)

Short answer

The integral by interpreting it in terms of areas is \(2.5\)

Step-by-step solution

Concept of definite integral in terms of area

A definite integral can be interpreted as a net area, that is, a difference of areas: \(\int_{{J_a}}^b f (x)dx = {A_1} - {A_2}\).

Property of definite integral is \(\int_a^c f (x)dx + \int_c^b f (x)dx = \int_a^b f (x)dx\)

 Step 2: Integrate the given integral

The given integral is \(\int_{ - 1}^2 | x|dx\)

The integral is the area between the curve \(y = |x|\) and the \(x\)-axis between \(x = - 1\) and \(x = 2\). By "interpreting it in terms of areas" they mean for us to look at the graph and use the basic geometric formulas that we know from long ago.

using the area of the triangle formula

There are two triangles, so we can use the triangle area formula

\(A = \frac{{( base )( height )}}{2}\)

Also, since both triangles are above the\(x\)-axis, the areas are completely positive.

\(\int_{ - 1}^2 | x|dx = \frac{{(1)(1)}}{2} + \frac{{(2)(2)}}{2} = \frac{5}{2}\)

Hence, the integral by interpreting it in terms of areas is \(2.5\).