INTEGRATION 04

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation with accumulation.

The main idea

Differentiation and integration appear to do different things.

Differentiation finds a rate of change. Integration accumulates a quantity over an interval.

The Fundamental Theorem of Calculus shows that these two processes are deeply connected.

\[ \boxed{ \text{differentiation} \longleftrightarrow \text{integration} } \]

1. Start with accumulation

Suppose \(f\) is continuous and define

\[ A(x) = \int_a^x f(t)\,dt. \]

The function \(A(x)\) gives the accumulated value of \(f\) from the fixed point \(a\) up to the changing point \(x\).

2. What happens when \(x\) changes?

Increase \(x\) by a small amount \(h\).

Then

\[ A(x+h) = \int_a^{x+h}f(t)\,dt. \]

The change in accumulated area is

\[ A(x+h)-A(x). \]

This is exactly the accumulation over the short interval from \(x\) to \(x+h\):

\[ A(x+h)-A(x) = \int_x^{x+h}f(t)\,dt. \]

3. Approximate the small accumulated area

When \(h\) is small, \(f(t)\) is close to \(f(x)\) throughout the short interval.

So the added area is approximately

\[ f(x)h. \]

Therefore,

\[ A(x+h)-A(x) \approx f(x)h. \]

Divide by \(h\):

\[ \frac{A(x+h)-A(x)}{h} \approx f(x). \]

4. Let the interval shrink

\[ h\to0, \]

the approximation becomes exact.

Therefore,

\[ \boxed{ A'(x)=f(x) } \]

where

\[ A(x)=\int_a^x f(t)\,dt. \]

So differentiating an accumulation function returns the original function.

5. First form of the theorem

\[ A(x)=\int_a^x f(t)\,dt, \]

then

\[ \boxed{ \frac{d}{dx} \left( \int_a^x f(t)\,dt \right) = f(x) } \]

provided \(f\) is continuous.

6. Why the variable \(t\) appears

\[ \int_a^x f(t)\,dt, \]

\(t\) is a temporary variable used inside the integral.

The final result depends on the upper limit \(x\).

We could equally write

\[ \int_a^x f(u)\,du. \]

The symbols \(t\) and \(u\) play the same role here.

7. Example

Let

\[ A(x)=\int_0^x t^2\,dt. \]

The theorem immediately gives

\[ A'(x)=x^2. \]

We can also calculate \(A(x)\):

\[ A(x) = \left[ \frac{t^3}{3} \right]_0^x = \frac{x^3}{3}. \]

Differentiating,

\[ A'(x)=x^2. \]

The two approaches agree.

8. From accumulation to antiderivatives

Suppose

\[ F'(x)=f(x). \]

Then \(F\) is an antiderivative of \(f\).

The Fundamental Theorem tells us that accumulated change between \(a\) and \(b\) is

\[ \boxed{ \int_a^b f(x)\,dx = F(b)-F(a) } \]

This is the second major form of the theorem.

9. Why this is so powerful

The definite integral was originally defined through accumulation and limits of sums.

But the theorem allows us to calculate it using an antiderivative.

\[ \text{many tiny contributions} \] \[ \downarrow \] \[ \int_a^b f(x)\,dx \] \[ \downarrow \] \[ \boxed{ F(b)-F(a) } \]

So we do not need to calculate infinitely many rectangles directly.

10. Example

Evaluate

\[ \int_1^3 2x\,dx. \]

An antiderivative is

\[ F(x)=x^2. \]

Therefore,

\[ \int_1^3 2x\,dx = F(3)-F(1). \]

\[ =9-1 \] \[ =8. \]

Hence,

\[ \boxed{ \int_1^3 2x\,dx=8 } \]

11. Rate and total change

Suppose

\[ \frac{dy}{dx}=f(x). \]

Integrating from \(a\) to \(b\),

\[ \int_a^b \frac{dy}{dx}\,dx = \int_a^b f(x)\,dx. \]

The left side gives the total change in \(y\):

\[ \boxed{ y(b)-y(a) = \int_a^b f(x)\,dx } \]

This is one of the most useful interpretations of the theorem.

12. Motion example

If velocity is

\[ v(t)=\frac{ds}{dt}, \]

then accumulated displacement from \(t=a\) to \(t=b\) is

\[ \int_a^b v(t)\,dt. \]

Therefore,

\[ \boxed{ s(b)-s(a) = \int_a^b v(t)\,dt } \]

Differentiation gives velocity from position. Integration recovers displacement from velocity.

13. Visual meaning

When \(x\) moves slightly to the right, a thin new strip of area is added.

The rate at which accumulated area grows is exactly the height \(f(x)\).

14. The two central statements

\[ \boxed{ \frac{d}{dx} \left( \int_a^x f(t)\,dt \right) = f(x) } \] \[ \boxed{ \int_a^b f(x)\,dx = F(b)-F(a), \qquad F'(x)=f(x) } \]

The first says that differentiation reverses accumulation.

The second says that accumulated change can be calculated using an antiderivative.

15. The full connection

\[ \boxed{ \text{function} \xrightarrow{\text{differentiate}} \text{rate} } \] \[ \boxed{ \text{rate} \xrightarrow{\text{integrate}} \text{accumulated change} } \]

The Fundamental Theorem of Calculus is what makes these two parts of calculus one coherent subject.