Question

Is integration the opposite of differentiation? In what sense do derivatives “undo” integrals? In what sense do integrals “undo” derivatives? In what sense do they not? In your answer, be sure to consider indefinite integrals, definite integrals, and accumulation functions defined by integrals

Short answer

Yes. Differentiation is the antithesis of integration.

Step-by-step solution

Given information

The second calculus fundamental theorem and the area accumulation functions

Calculation.

The area accumulation function ${\int }_a^{b}f\left(x\right)dx$should be taken into consideration if function$f$is continuous on the intervallocalid="1661254656606" $[\mathrm{a},\mathrm{b}]$.

The definite ${\int }_a^{b}f\left(x\right)dx$defines the signed region between the graph of $f$and the localid="1661254666018" $x$-axis.

Using the differentiation function of any definite or indefinite integral, as will be described below, any integral can be undone. Similar to how any differentiation can be reversed using its integral, which is described below, thus there is a connection between these.

If localid="1663919734077" $f\text{is continuous on the interval}[a,b]$, where localid="1658495130972">F(x)=∫abf(x)dxcontinuous on $[a,b]$and differentiable on $(a,b)$the function localid="1661254709048" $F$is anti derivative of f, that is,$F^{\text{'}}\left(x\right)=f\left(x\right)$. This is called the second fundamental theorem of calculus.

- If localid="1663919802653" role="math" $f\mathrm{is}\mathrm{continuous},\mathrm{where}F\left(x\right)+\mathrm{C}=\int f\left(x\right)dx$is continuous and differentiable. The function localid="1661254721854" $F$is anti derivative of f, that is $F^{\text{'}}\left(x\right)=f\left(x\right)$.

Example for the given statement.

Let us consider the indefinite integral $\int x^{2}dx$.

Apply the theory

$$\begin{gathered} \frac{dF\left(x\right)}{dx}=f\left(x\right) \\ \int f\left(x\right)dx=F\left(x\right)+C \end{gathered}$$

Then,

$$\begin{gathered} F\left(x\right)=x^3 \\ \frac{1}{3}·\frac{d{x}^3}{dx}=x^2=f\left(x\right) \\ \int x^{2}dx=\frac{x^3}{3}+C \end{gathered}$$

So, yes. Differentiation is the antithesis of integration.