Fundamental Theorem of Calculus

Introduction

The fundamental theorem of calculus is a theorem that links the concepts derivative and integral a function.

The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say $F$, of some function $f$ may be obtained as the integral of f with a variable bound of integration. This implies the existence of antiderivatives for continuous functions [1, p. 205].

Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function $f$ over some interval can be computed by using any one, say $F$, of its infinitely many antiderivatives. This part of the theorem has key practical applications, because explicitly finding the antiderivative of a function by symbolic integration allows for avoiding numerical integration to compute integrals.

Simulation

In the following simulation, the top graph shows the function $f(t)$ and the value of the definite integral for each upper limit $x$, with lower limit $a$. The definite integral is represented with the shaded region between the graph of the function and the $x$-axis.

The bottom graph shows the accumulation function $$A(x)=\int_a^xf(t)dt$$ for each upper limit $x$, with lower limit $a$.

Things to try:

• Select an option, at the bottom, to explore the Accumulation function or the Derivative of the accumulation function.
• Drag point $x$ along the $x$-axis in the top graph to observe the relationship between the two graphs.
• Drag the point $a$ to change its value.
• Change the function $f(t)$. Remark: $f$ must be continuous.
   Example 1: sin(t)
   Example 2: t^2-1/2t+1
   Example 3: 1/2abs(t)

Worksheet exemplar

The following file contains activities and problems associated with the simulation.

• Fundamental Theorem of Calculus (pdf)
• Fundamental Theorem of Calculus (tex)

References

[1] Ghorpade, S. R. & Limaye, B. V. (2006). A Course in Calculus and Real Analysis. Springer Science+Business Media. New York.