9. (a) \( \int_{1}^{2} 2 x d x \)

To solve the integral \( \int_{1}^{2} 2x \, dx \), we first need to find the antiderivative of \( 2x \). The antiderivative is \( x^2 \). Now, we can evaluate it from 1 to 2: \[ \left[ x^2 \right]_{1}^{2} = (2^2) - (1^2) = 4 - 1 = 3. \] Thus, the result of the integral is 3. As a fun fact, did you know that integral calculus has its roots in the study of motion? Historically, mathematicians like Newton and Leibniz were developing the concepts of limits and accumulation of quantities, leading to the idea of the definite integral, which helps us calculate areas under curves—like the area you'd find between the line \( y = 2x \) and the x-axis from 1 to 2! For practical applications, understanding definite integrals is crucial in fields like physics and engineering. For example, calculating the distance traveled by an object over time can be achieved by integrating the velocity function. In this case, if you have a speed of \( 2x \) m/s between points 1 and 2 seconds, integrating gives you the total distance covered, which helps in designing efficient transportation systems or analyzing motion dynamics!