To solve the integral

$\int_{-3}^{0} \left(1 + \sqrt{9 - x^2}\right) \, dx$

we can break down the solution into manageable steps.

Step 1: Split the Integral

We can separate the integral into two parts:

$\int_{-3}^{0} \left(1 + \sqrt{9 - x^2}\right) \, dx = \int_{-3}^{0} 1 \, dx + \int_{-3}^{0} \sqrt{9 - x^2} \, dx$

So, we have:

1. $\int_{-3}^{0} 1 \, dx$
2. $\int_{-3}^{0} \sqrt{9 - x^2} \, dx$

Step 2: Evaluate the First Integral

The first integral, $\int_{-3}^{0} 1 \, dx$, is straightforward:

$\int_{-3}^{0} 1 \, dx = \left[ x \right]_{-3}^{0} = 0 - (-3) = 3$

Step 3: Evaluate the Second Integral

The second integral, $\int_{-3}^{0} \sqrt{9 - x^2} \, dx$, resembles the formula for the area of a semicircle. The function $\sqrt{9 - x^2}$ represents the upper half of a circle with radius $3$ (since $9 = 3^2$).

The area of a semicircle of radius $3$ is given by $\frac{1}{2} \pi r^2$. So, for this part:

$\int_{-3}^{3} \sqrt{9 - x^2} \, dx = \frac{1}{2} \pi \cdot 3^2 = \frac{9\pi}{2}$

However, since our integral is from $-3$ to $0$ (covering only half of the semicircle), we take half of $\frac{9\pi}{2}$:

$\int_{-3}^{0} \sqrt{9 - x^2} \, dx = \frac{9\pi}{4}$

Step 4: Combine the Results

Now, we add the two parts together:

$\int_{-3}^{0} \left(1 + \sqrt{9 - x^2}\right) \, dx = 3 + \frac{9\pi}{4}$

Final Answer

$\int_{-3}^{0} \left(1 + \sqrt{9 - x^2}\right) \, dx = 3 + \frac{9\pi}{4}$

Would you like more details on any step, or do you have any questions?

1. How would this change if the integral limits were from $-3$ to $3$?
2. What is the geometrical interpretation of the term $\sqrt{9 - x^2}$ in this integral?
3. How would the approach differ if the term under the square root were not a perfect square, such as $\sqrt{8 - x^2}$?
4. Can we apply substitution techniques to evaluate integrals of this form? If so, which ones?
5. How would the solution change if the integrand were $1 - \sqrt{9 - x^2}$ instead?

Tip: For integrals involving $\sqrt{a^2 - x^2}$, consider the geometric properties of circles or trigonometric substitution methods.