First Fundamental Theorem of Integral Calculus

Bernoulli’s formula

Solid of revolution about a fixed axis means

Second Fundamental Theorem of Integral Calculus

First Fundamental Theorem and Second Fundamental Theorem
of Integral Calculus

If f (x) be a continuous function defined on a closed interval [a,b]and F(x)= ∫axf(u)du,a< x<b then, ddx F(x) = f (x). In other words, F(x) is an anti-derivative of f (x).

An extension of the formula of integration by parts

A solid is generated when a plane region in a given plane
undergoes one full revolution about a fixed axis in the plane

If f (x) be a continuous function defined on a closed interval [a,b] and F(x) is an
anti-derivative of f (x), then,
∫abf(x)dx= F(b) -F (a)

Establishs the connection between a function and its
anti-derivative (if it exists)