Q. 5 - Calculus

Question

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Step-by-Step Solution

Answer

Differences between the definite and indefinite integrals are following.

1. Indefinite integral concern with the antiderivative whereas definite integral concern with the area under the graph and x-axis.
2. The indefinite integral is a family of antiderivative differed by a constant whereas the definite integral is a number.
3. The indefinite integral is determined by antidifferentiation whereas the definite integral is determined by using Reimann sums.

Step 1. Given information.

The given topics on which we have to differentiate are the definite and indefinite integrals.

Step 2. definite and indefinite integrals.

The definite integral of a function from $x=a \mathrm{to} x=b$is defined by a number as follows.

$$\begin{gathered} {\int }_a^{b}f\left(x\right) dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left({\stackrel{*}{x}}_k\right) ∆x \\ ∆x=\frac{b-a}{n} and x_k=a+k∆x \end{gathered}$$

The indefinite integral of a continuous function f is a family of antiderivative differ by a constant.

${\int }_{}f\left(x\right) dx=F\left(x\right)+c$

Step 3. Differences.

Differences between the definite and indefinite integrals are following.

1. Indefinite integral concern with the antiderivative whereas definite integral concern with the area under the graph and x-axis.
2. The indefinite integral is a family of antiderivative differed by a constant whereas the definite integral is a number.
3. The indefinite integral is determined by antidifferentiation whereas the definite integral is determined by using Reimann sums.