How does specifying the initial value $A(0)=0$ select the unique antiderivative $x^3/3$ from the general family $x^3/3 + C$?

Conditions

• Integrand is $f(t) = t^2$
• Lower limit of integration is 0
• Accumulation starts from zero

Reasoning, step by step

1. Find the general antiderivative of $x^2$: $\int x^2 dx = \frac{x^3}{3} + C$.
2. Define the specific accumulation function $A(x) = \int_0^x t^2 dt$.
3. Evaluate $A(0)$ based on the definition: the integral from 0 to 0 is 0.
4. Set the general antiderivative equal to 0 at $x=0$: $\frac{0^3}{3} + C = 0$.
5. Solve for $C$: $C=0$.
6. Write the final specific function: $A(x) = \frac{x^3}{3}$.

Example

The script concludes: 'Antiderivatives may differ by constants. For accumulation of x² from zero, $A(0)=0$ selects $A(x)=x$³/3 from the family x³/3+C.'

Common misconceptions

• Assuming that $\int x^2 dx$ automatically equals $x^3/3$ without considering the constant.
• Confusing the indefinite integral notation with the definite accumulation function.

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Connected concepts

Explore next

• What would the function be if the lower limit was $a \neq 0$?
• How does this illustrate the connection between the two parts of the FTC?

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