Calculating Areas Under and Between Curves Using Integral Calculus

Integral calculus is a fundamental mathematical tool for calculating areas bounded by curves and axes or between two curves, using definite integrals. The area under a curve $y=f(x)$ from $x=a$ to $x=b$ is computed by evaluating the definite integral $\displaystyle \int_a^b f(x) \, dx$, which gives the exact area when is positive and continuous over the interval. When determining the area between two curves and where on , the formula applies. If the curves intersect within the interval, the integral must be split at intersection points to maintain correct subtraction order. Similarly, areas can be calculated with respect to the y-axis by integrating functions expressed as in terms of . The geometric interpretation of definite integrals as net area supports applications in engineering fields such as displacement calculation, work done, volume determination, and stress analysis. Accurately partitioning intervals and understanding the integral formulations provide precision in solving engineering design and optimization problems.