Let's work through these two integrals step by step.

Part a:

We are asked to evaluate the double integral:

$\int_0^2 \int_0^{2\sqrt{4-x^2}} \left(4 - x^2 - \frac{1}{4} y^2\right) \, dy \, dx$

1. Set up the outer and inner integrals:

   • The outer integral runs with respect to $x$ from $0$ to $2$.
   • The inner integral is with respect to $y$, with limits from $0$ to $2\sqrt{4 - x^2}$.

2. First, solve the inner integral:

$\int_0^{2\sqrt{4-x^2}} \left(4 - x^2 - \frac{1}{4} y^2\right) dy$

The integral of each term inside the integrand is as follows:

• $\int 4 \, dy = 4y$,
• $\int -x^2 \, dy = -x^2 y$,
• $\int -\frac{1}{4}y^2 \, dy = -\frac{1}{12} y^3$.

Now substitute the upper and lower limits of $y = 2\sqrt{4 - x^2}$ into the result and then integrate with respect to $x$.

Part b:

We are asked to evaluate the triple integral:

$\int_0^1 \int_0^{z^2} \int_0^3 y \cdot \cos^5(z) \, dx \, dy \, dz$

1. Set up the outer, middle, and inner integrals:

   • The outer integral is with respect to $z$ from $0$ to $1$.
   • The middle integral is with respect to $y$ from $0$ to $z^2$.
   • The inner integral is with respect to $x$ from $0$ to $3$.

2. Solve the innermost integral: Since there are no $x$-terms in the integrand, the inner integral just gives a factor of $3$ (since $\int_0^3 dx = 3$).

Thus, the integral becomes:

$3 \int_0^1 \int_0^{z^2} y \cdot \cos^5(z) \, dy \, dz$

3. Solve the middle integral: The integral of $y$ is straightforward:

$\int_0^{z^2} y \, dy = \frac{1}{2} z^4$

Thus, the expression becomes:

$\frac{3}{2} \int_0^1 z^4 \cdot \cos^5(z) \, dz$

Finally, evaluate this integral numerically or by using integration techniques suitable for powers of trigonometric functions.

Would you like more details on any of these steps? Here are 5 related questions for further practice:

1. Can you evaluate the first integral with exact limits for $y$ in part (a)?
2. How do you handle trigonometric integrals like $\cos^5(z)$ in part (b)?
3. What are the geometrical interpretations of these double and triple integrals?
4. What techniques can be used for simplifying integrals with square roots in the limits?
5. How does the order of integration affect the outcome of the integrals?

Tip: Breaking down complex integrals into smaller parts and solving step by step simplifies the process.