To evaluate the definite integral:

$\int_{0}^{6} x \, dx$

Step 1: Find the Antiderivative

The antiderivative of $x$ is:

$\frac{x^2}{2}$

Step 2: Evaluate at the Limits

Now, we apply the limits $0$ and $6$:

$\left[ \frac{x^2}{2} \right]_{0}^{6}$

$= \frac{6^2}{2} - \frac{0^2}{2}$

$= \frac{36}{2} - \frac{0}{2}$

$= 18 - 0$

$= 18$

Final Answer:

$18$

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Related Questions:

1. How do you find the antiderivative of $x^n$?
2. What are the properties of definite integrals?
3. How do you interpret an integral geometrically?
4. How do definite and indefinite integrals differ?
5. What happens if the limits of integration are reversed?

Tip:

The definite integral $\int_a^b f(x)dx$ represents the area under the curve $y = f(x)$ from $x = a$ to $x = b$.