Solutions:

Part (a)

Evaluate the integral:

$\int_0^2 (1 + x^2) \, dx$

1. Split the integral: $\int_0^2 (1 + x^2) \, dx = \int_0^2 1 \, dx + \int_0^2 x^2 \, dx$

2. Solve each term:

   • $\int_0^2 1 \, dx = [x]_0^2 = 2 - 0 = 2$
   • $\int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$

3. Combine results: $\int_0^2 (1 + x^2) \, dx = 2 + \frac{8}{3} = \frac{6}{3} + \frac{8}{3} = \frac{14}{3}$

Part (b)

Evaluate the integral:

$\int_0^{\sqrt{x}} (1 + 3x^2) \, dx$

Let’s clarify the limits: Are you asking to integrate $\int_0^{\sqrt{y}} (1 + 3x^2) \, dx$ with respect to $x$? If so, the solution requires detailed substitution. Confirm before I delve into the ultimate steps.

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Would you like me to explain the steps further? Here are five related questions to expand on this:

1. How does splitting an integral help simplify the solution process?
2. Can you explain the geometric interpretation of definite integrals?
3. How do substitution techniques work for more complex integrals?
4. What happens if the limits of integration are swapped?
5. Why is it important to carefully analyze the integral's limits and variables?

Tip: Always double-check the limits and variable of integration to avoid errors in interpretation!