# The Fundamental Theorem of Calculus

## The Fundamental Theorem of Calculus (FTC), Part 1

• This theorem has such a big name, because it is a big theorem. It gives a direct connection between two seemingly unrelated topics:
• derivatives, that is slopes of graphs,
• and definite integrals, that is net areas under graphs.
• Theorem. Let $$f$$ be a continuous function on $$[a,b]$$. Then the function $$g$$ defined by $g(x)=\int_a^xf(t)\,\mathrm dt\quad a\le x\le b$ is continuous on $$[a,b]$$, differentiable on $$(a,b)$$, and satisfies $g'(x)=f(x).$
• That is, we have a formula between definite integrals and antiderivatives!
• Note that this implies that if $$F$$ is any antiderivative of $$f$$, then we have $F(x)=\int_a^xf(t)\,\mathrm dt+C\quad a\le x\le b$ for some constant $$C$$.
• Warning. You need to make sure that the upper boundary $$x$$ is a different variable from the variable of integration $$t$$. You get wrong answers if you miss this!
• You can see the proof of this theorem in the textbook.

## Exercises

• Since we started considering functions where the function variable (usually $$x$$) is a boundary of an integral, we want to make sense of integrals $$\int_b^af(x)\,\mathrm dx$$ in case $$a<b$$.
• We define $$\int_b^af(x)\,\mathrm dx$$ as $$-\int_a^bf(x)\,\mathrm dx$$.
• The interval splitting formula $$\int_a^bf(x)\,\mathrm dx=\int_a^cf(x)\,\mathrm dx+\int_c^bf(x)\,\mathrm dx$$ remains true for $$a,b,c$$ not necessarily satisfying $$a\le c\le b$$.
• Exercises. 5.3: 8, 12, 14, 18, 60, 62, 64, 65.

## FTC Part 2

• Theorem. Let $$f$$ be a continuous function on $$[a,b]$$. Suppose that $$F$$ is an antiderivative of $$f$$. Then we have $\int_a^bf(x)\,\mathrm dx=F(b)-F(a)$
• Usually this is the way we want to calculate definite integrals, so this is the main reason why knowing how to find antiderivatives of functions is important.
• Notation. Since because of this we see formulas like $$F(b)-F(a)$$ a lot, where $$F(x)$$ might itself have a long formula, we use the following notations. $F(x)\Big|_{x=a}^b=F(x)\Big|_a^b=F(x)\Big]_a^b=\Big[F(x)\Big]_a^b=F(x=b)-F(x=a).$
• Exercises. 5.3: 20, 22, 26, 32, 40, 42, 44, 52, 76, 82