Question

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

Short answer

The value of the integral \( \int_{0}^{b} x d x \) is \( \frac{b^2}{2} \).

Step-by-step solution

Understanding the Concept of Area Under the Curve

Firstly, we need to conceptualize the curve that the integral represents. The curve is \( y = x \). The area under this curve, from 0 to b, represents a triangle. This is because the line \( y = x \) is a straight line that passes through the origin forming a right-angled triangle under it.

Formula for Area

Remember, the area \( A \) of a triangle is given by the formula \( \frac{1}{2}bh \), where \( b \) is the base and \( h \) is the height of the triangle. Here, both base and height are equal to \( b \) (The interval 0 to b).

Calculate the Area

Apply the base and height in the area formula \( A = \frac{1}{2}bh \). With \( b \) as base and \( h \), we get \( A = \frac{1}{2}b \times b = \frac{b^2}{2} \). The resulting area represents the value of the definite integral.