Question

Prove that $$\int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2}$$

Short answer

Thus, \(∫_{a}^{b} x dx = \frac{b^{2}-a^{2}}{2}\).

Step-by-step solution

Find the indefinite integral

The indefinite integral of \(x\) is \(∫x dx = \frac{x^2}{2} + C\), where \(C\) is the constant of integration.

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that \(∫_{a}^{b} f(x) dx = F(b) - F(a)\), where \(F(x)\) is the antiderivative of \(f(x)\). So, \(∫_{a}^{b}xdx = \frac{b^2}{2} - \frac{a^2}{2}\).

Simplify the result

By combining the fraction, the result is \(\frac{b^2 - a^2}{2}\).