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

To compute the integral \( A = \int _ { 2 } ^ { 5 } (2x - 3) \, dx \), we start by finding the antiderivative of the function \( 2x - 3 \), which is \( x^2 - 3x \). Now we evaluate this from 2 to 5: \[ A = \left[ x^2 - 3x \right] _{2}^{5} = \left( 5^2 - 3 \cdot 5 \right) - \left( 2^2 - 3 \cdot 2 \right) = (25 - 15) - (4 - 6) = 10 + 2 = 12. \] So, the value of the integral \( A \) is 12. In the world of calculus, understanding the concept of integration allows us not just to find areas under curves but to delve into more complex applications like calculating total distance from velocity functions! It's fascinating how a simple equation can unlock the secrets of nature, from physics to economics. Don’t let the fear of making mistakes in integration loom over you! A common pitfall is neglecting to apply the limits correctly or mistakenly calculating the antiderivative. Always double-check your work and remember that practice makes perfect, so grab a stack of integrals and start integrating your way to mastery!