Let's solve the given integral step by step.

The integral is:

$\int_0^1 \left( x^2 - \frac{1}{3} \right) dx$

Step 1: Split the integral

We can split the integral into two parts: $\int_0^1 \left( x^2 - \frac{1}{3} \right) dx = \int_0^1 x^2 \, dx - \int_0^1 \frac{1}{3} \, dx$

Step 2: Solve each integral

1. First Integral: $\int_0^1 x^2 \, dx$:

   Use the power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ Here $n = 2$, so: $\int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}.$

2. Second Integral: $\int_0^1 \frac{1}{3} \, dx$:

   Since $\frac{1}{3}$ is a constant, the integral becomes: $\int_0^1 \frac{1}{3} \, dx = \frac{1}{3} \int_0^1 1 \, dx = \frac{1}{3} \left[ x \right]_0^1 = \frac{1}{3} \left( 1 - 0 \right) = \frac{1}{3}.$

Step 3: Combine the results

Now subtract the results of the two integrals: $\int_0^1 \left( x^2 - \frac{1}{3} \right) dx = \frac{1}{3} - \frac{1}{3} = 0.$

Final Answer:

The value of the integral is: $\boxed{0}$

Let me know if you need further clarification or have additional questions!

Here are 5 related questions to expand your understanding:

1. How do you integrate polynomials of higher degrees like $x^n$?
2. How would the result change if the limits of the integral were different?
3. What is the geometrical interpretation of this integral?
4. How do you calculate definite integrals involving trigonometric functions like $\sin x$ or $\cos x$?
5. What is the fundamental theorem of calculus, and how does it apply here?

Tip: Always check if an integral can be simplified by separating constants or breaking it into smaller parts!