\( A = \int _ { 1 } ^ { 5 } 2 x - 3 d x \)

To find the value of the integral \( A = \int _ { 1 } ^ { 5 } (2x - 3) \, dx \), we first need to compute the antiderivative. The antiderivative of \( 2x - 3 \) is \( x^2 - 3x \). Now we evaluate this from 1 to 5: \[ A = \left[ x^2 - 3x \right] _ { 1 } ^ { 5 } = \left( 5^2 - 3(5) \right) - \left( 1^2 - 3(1) \right) = (25 - 15) - (1 - 3) = 10 + 2 = 12. \] So, the value of the integral is \( A = 12 \). It's fascinating to note that calculus, particularly integration, has roots that stretch back over 300 years. Pioneers like Isaac Newton and Gottfried Wilhelm Leibniz developed the foundational theory, revolutionizing not just mathematics, but also physics and engineering, allowing for the precise calculation of areas, volumes, and other quantities, leading to advancements in countless fields. In real-world applications, integrals like this one are crucial for determining accumulated quantities, such as distance traveled over time or area under a curve representing speed. For instance, if the function \( 2x - 3 \) described a speed in meters per second over a given time interval, then the integral would represent the total distance traveled during that time—making it immensely useful in physics, economics, and engineering.