Let's solve each of the integrals given in the problem step-by-step.

Information Given

1. $f(x)$ is continuous on the interval $[-3, 3]$.
2. $\int_0^3 f(x) \, dx = 6$.

Integral Evaluations

(a) $\int_0^3 \left[ f(x) + 2 \right] \, dx$

Using the property of integrals, we can separate this integral: $\int_0^3 \left[ f(x) + 2 \right] \, dx = \int_0^3 f(x) \, dx + \int_0^3 2 \, dx$

From the given information, we know: $\int_0^3 f(x) \, dx = 6$

Now, calculate $\int_0^3 2 \, dx$: $\int_0^3 2 \, dx = 2 \cdot (3 - 0) = 6$

Therefore: $\int_0^3 \left[ f(x) + 2 \right] \, dx = 6 + 6 = 12$

Answer for (a): 12

(b) $\int_{-2}^1 \left( f(x) + 2 \right) \, dx$

Similarly, separate the integral: $\int_{-2}^1 \left( f(x) + 2 \right) \, dx = \int_{-2}^1 f(x) \, dx + \int_{-2}^1 2 \, dx$

Since we do not have information about $\int_{-2}^1 f(x) \, dx$, we cannot directly calculate this term. This integral cannot be fully evaluated with the given information.

(c) $\int_{-3}^3 f(x) \, dx$ (if $f$ is even)

For an even function $f(x)$, we know that: $\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx$

Given $\int_0^3 f(x) \, dx = 6$, it follows: $\int_{-3}^3 f(x) \, dx = 2 \cdot 6 = 12$

Answer for (c): 12

(d) $\int_{-3}^3 f(x) \, dx$ (if $f$ is odd)

For an odd function $f(x)$, we have: $\int_{-a}^a f(x) \, dx = 0$

Therefore: $\int_{-3}^3 f(x) \, dx = 0$

Answer for (d): 0

Summary of Answers

1. (a) $\int_0^3 [f(x) + 2] \, dx = 12$
2. (b) $\int_{-2}^1 (f(x) + 2) \, dx$: Cannot be determined from the given information.
3. (c) $\int_{-3}^3 f(x) \, dx = 12$ (if $f$ is even)
4. (d) $\int_{-3}^3 f(x) \, dx = 0$ (if $f$ is odd)

Would you like further details on any specific part of these calculations, or do you have questions on the properties used?

Additional Questions

1. How would the result change in (b) if more information about $f(x)$ on $[-2, 1]$ was given?
2. What if we knew $f(x)$ was neither even nor odd? How would that affect (c) and (d)?
3. Can we compute $\int_{-3}^3 f(x) + 2 \, dx$ with the given information?
4. How does the linearity property of integrals simplify problems like (a)?
5. What are the implications of continuity for $f(x)$ over the interval $[-3, 3]$?

Tip

For functions defined over symmetric intervals like $[-a, a]$, always check if they are even or odd. This can simplify calculations significantly.