Question

In Exercises \(23-28,\) use areas to evaluate the integral. $$\int_{a}^{2 a} x d x, \quad a>0$$

Short answer

The value of the integral \(\int_{a}^{2 a} x d x\) is \(\frac{3}{2}a^2\).

Step-by-step solution

Write the integrand as notation of a function

One can write the object of integration \(x\) as a function notation \(f(x) = x\).

Draw function

Draw the function \(f(x)=x\). This is a straight line passing through the origin under a 45 degrees angle.

Identify region of integration

Identify the region under the curve of \(f(x) = x\) from \(x = a\) to \(x = 2a\). This forms a trapezoid under the line.

Calculate area of the defined region

The area under the curve of \(f(x) = x\) from \(x = a\) to \(x = 2a\) can be obtained as the area of a trapezoid. This is denoted as \(A = \frac{1}{2}(a+2a)(2a - a) = \frac{3}{2}a^2\).

Evaluate the integral

Use the calculated area as the value of the definite integral. Thus, \(\int_{a}^{2 a} x d x = \frac{3}{2}a^2\).