Let's evaluate the definite integrals by analyzing the given graph of $f(x)$, which consists of line segments and a semicircle. We'll use geometric formulas to compute the areas under the curve (considering signs for areas below the x-axis).

Analysis:

(a) $\int_0^2 f(x) \, dx$:

This corresponds to the area under the semicircle from $x = 0$ to $x = 2$, below the x-axis. The semicircle has radius 2, and its area is $-\frac{1}{2} \pi r^2$ because it is below the x-axis. $\text{Area} = -\frac{1}{2} \pi (2)^2 = -2\pi.$

(b) $\int_2^6 f(x) \, dx$:

This corresponds to the area of the triangle from $x = 2$ to $x = 6$, which lies above the x-axis. The triangle's base is 4 (from $x = 2$ to $x = 6$), and its height is 2. $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 2 = 4.$

(c) $\int_{-4}^2 f(x) \, dx$:

This involves two regions:

1. From $x = -4$ to $x = 0$ (the semicircle below the x-axis, as in part (a)).
2. From $x = 0$ to $x = 2$ (also the semicircle below the x-axis, as computed earlier). The total area is: $\text{Total area} = -2\pi \, (\text{left semicircle from } -4 \text{ to } 0) + (-2\pi \, \text{right semicircle from } 0 \text{ to } 2).$ $\text{Total area} = -4\pi.$

(d) $\int_{-4}^6 f(x) \, dx$:

Combine areas:

1. The semicircle from $x = -4$ to $x = 2$ contributes $-4\pi$ (from part (c)).
2. The triangle from $x = 2$ to $x = 6$ contributes $4$ (from part (b)). $\text{Total area} = -4\pi + 4.$

(e) $\int_{-4}^6 |f(x)| \, dx$:

Here, we sum the absolute values of all areas (ignoring whether they are above or below the x-axis):

1. The semicircle (both halves) contributes $(4\pi)$.
2. The triangle contributes $4$. $\text{Total area} = 4\pi + 4.$

(f) $\int_{-4}^6 [f(x) + 2] \, dx$:

Using the property of integrals: $\int_{-4}^6 [f(x) + 2] \, dx = \int_{-4}^6 f(x) \, dx + \int_{-4}^6 2 \, dx.$

1. $\int_{-4}^6 f(x) \, dx = -4\pi + 4$ (from part (d)).
2. $\int_{-4}^6 2 \, dx = 2 \cdot (6 - (-4)) = 20$. $\text{Total area} = (-4\pi + 4) + 20 = -4\pi + 24.$

Correct Answers:

• (a) $-2\pi$,
• (b) $4$,
• (c) $-4\pi$,
• (d) $-4\pi + 4$,
• (e) $4\pi + 4$,
• (f) $-4\pi + 24$.

Would you like further clarifications on any of these?

Related Questions:

1. What is the general process for computing definite integrals using geometric shapes?
2. How does the sign of the integral change based on the position of the curve relative to the x-axis?
3. How do we use symmetry in definite integrals for functions like semicircles?
4. Why does adding a constant to $f(x)$ modify the integral in the way seen in part (f)?
5. What happens to the definite integral if we reverse the limits of integration?

Tip:

For semicircles, always confirm the radius and use $\pi r^2$ appropriately (with half the area if it’s a semicircle).