B - Calculus

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. 

Step-by-Step Solution

Answer

Yes. The integration is the opposite of differentiation.

Step1: Given Information

As we know,

• The area accumulation functions
• The second fundamental theorem of calculus 

Consider the area accumulation function${\int }_a^{b}f\left(x\right)dx$

• If function $f$ is continuous on the interval $[a, b]$

The signed area between the graph of$f$ and the x-axis is given by the definite ${\int }_a^{b}f\left(x\right)dx$

• Any definite integral or indefinite integral can be undone using it's differentiation function as explained below. Same way any differentiation can be undone using it's s integral as explained below. since the relationship between these is connected.
• If $f_{\text{is continuous on the interval }}[a,b]$, where $F\left(x\right)={\int }_a^{b}f\left(x\right)dx$ continuous on $[a,b]$ and differentiable on $(a, b)$ the function $F$ 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 $f_{\text{is continuous, where }}F\left(x\right)+\mathrm{C}=\int f\left(x\right)dx$ is continuous and differentiable. The function $F$ is anti derivative of $f$, that is $F^{\text{'}}\left(x\right)=f\left(x\right)$.

Step2: Let's Take the example

Consider the indefinite integral $\int x^{2}dx$

Apply the theory

$\frac{dF\left(x\right)}{dx}=f\left(x\right)$

$\int f\left(x\right)dx=F\left(x\right)+C$

Then,

$F\left(x\right)=x^3$

$\frac{1}{3}·\frac{d{x}^3}{dx}=x^2=f\left(x\right)$

$\int x^{2}dx=\frac{x^3}{3}+C$

Thus, Yes. The integration is the opposite of differentiation.