The Fundamental Theorem of Calculus (FTC)
The Fundamental Theorem of Calculus has two main parts:
First Part: If a function ( f ) is smooth and doesn't have any breaks between two points ( a ) and ( b ), and if ( F ) is a function that goes backward from ( f ) (we call this an antiderivative), then we can write:
[ \int_a^b f(x) , dx = F(b) - F(a) ]
This means the total area under the curve of ( f ) from point ( a ) to point ( b ) is found by taking the difference between ( F ) at point ( b ) and ( F ) at point ( a ).
Second Part: If ( f ) is still smooth between points ( a ) and ( b ), then we can create a new function ( F ) from ( f ) like this:
[ F(x) = \int_a^x f(t) , dt ]
This new function ( F ) is continuous (no breaks) between ( a ) and ( b ). It can also change smoothly without any sudden jumps.
Plus, if we look at the change of ( F ) at any point ( x ) within ( a ) and ( b ), we find that the slope or rate of change (which we call the derivative) matches the original function ( f(x) ). This can be written as:
[ F'(x) = f(x) ]
In simple terms, this theorem connects two big ideas in calculus: differentiation (finding the slope) and integration (finding the area). It shows us that we can use one to undo the other.
The Fundamental Theorem of Calculus (FTC)
The Fundamental Theorem of Calculus has two main parts:
First Part: If a function ( f ) is smooth and doesn't have any breaks between two points ( a ) and ( b ), and if ( F ) is a function that goes backward from ( f ) (we call this an antiderivative), then we can write:
[ \int_a^b f(x) , dx = F(b) - F(a) ]
This means the total area under the curve of ( f ) from point ( a ) to point ( b ) is found by taking the difference between ( F ) at point ( b ) and ( F ) at point ( a ).
Second Part: If ( f ) is still smooth between points ( a ) and ( b ), then we can create a new function ( F ) from ( f ) like this:
[ F(x) = \int_a^x f(t) , dt ]
This new function ( F ) is continuous (no breaks) between ( a ) and ( b ). It can also change smoothly without any sudden jumps.
Plus, if we look at the change of ( F ) at any point ( x ) within ( a ) and ( b ), we find that the slope or rate of change (which we call the derivative) matches the original function ( f(x) ). This can be written as:
[ F'(x) = f(x) ]
In simple terms, this theorem connects two big ideas in calculus: differentiation (finding the slope) and integration (finding the area). It shows us that we can use one to undo the other.