The process of finding the area under simple curves, such as linear and quadratic functions, is a fundamental application of integrals. This concept allows us to calculate the exact area enclosed by a curve between two points on the x-axis.

To find the area under a linear function, we can utilize the formula A = (1/2)bh, where A represents the area, b is the base or the interval of x-values, and h is the height or the respective y-value of the function. For example, to find the area under the line y = 2x + 3 between x = 1 and x = 4, we can substitute the given values into the formula to obtain A = (1/2)(3)(6) = 9 square units.

Similarly, for quadratic functions, we can integrate the function within the given interval to calculate the enclosed area. For instance, to find the area under the curve y = x^2 between x = 0 and x = 2, we can use the definite integral ∫(0 to 2) x^2 dx. Solving this integral yields an area of 8/3 square units.

By understanding how to find the area under simple curves, we can apply this knowledge to more complex functions and solve real-world problems where area calculations are crucial.