definite integral

The definite integral of \( f(x) \) from \( a \) to \( b \) represents the signed area between the graph of \( f(x) \), the x-axis, and the lines \( x = a \) and \( x = b \) and is given by:

\[
\int_{a}^{b} f(x) \, dx
\]

The Fundamental Theorem of Calculus, establishing the precise link between differentiation and definite integration.

The Mean Value Theorem for Integrals states that a continuous function attains its average value at least once over a closed interval.

The shell method is a technique for calculating the volume of a solid of revolution by integrating the volumes of thin cylindrical shells.

Volumes from parallel cross-sections are volumes of solids calculated by integrating the areas of their parallel cross-sections along a given axis.

The washer method is a technique for calculating the volume of a solid of revolution using definite integrals and annular cross-sections.

The surface area of revolution is the area of the surface generated by rotating a plane curve around a fixed axis.