Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves

Definite integrals are a cornerstone of calculus, allowing you to calculate the exact accumulation of a quantity over an interval. In A-Level Mathematics, you will learn to evaluate definite integrals using the Fundamental Theorem of Calculus, which links the integral of a function to its antiderivative. This topic is essential for solving real-world problems involving areas, volumes, and total change, and it builds directly on your knowledge of indefinite integration and differentiation.

The most common application of definite integrals is finding the area under a curve. By integrating a function between two limits, you obtain the signed area between the curve and the x-axis. You must be careful with signs: areas below the x-axis yield negative values, so you may need to split the integral into parts where the function is positive and negative. This skill is crucial for more advanced topics like volumes of revolution and in physics for work and displacement calculations.

Extending this, you can find the area between two curves by integrating the difference of the functions. The key is to identify which function is on top (greater y-value) over the interval of interest. This often requires solving equations to find intersection points, which become the limits of integration. Mastering these techniques prepares you for problems involving compound areas and sets the stage for integration in multiple dimensions.