Let's evaluate each integral by interpreting the graph of $f(x)$ in terms of areas.

The graph consists of geometric shapes: rectangles and triangles. We'll calculate the areas and keep in mind that areas below the x-axis are negative.

(a) $\int_0^4 f(x) \, dx$

From $x = 0$ to $x = 4$, the graph forms a triangle with a base of $4$ and height $4$.

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8$

Since the triangle is above the x-axis, the integral is positive.

Answer: $8$

(b) $\int_0^{10} f(x) \, dx$

Here, the interval splits into two regions:

• From $x = 0$ to $4$, we calculated an area of $8$ (above the x-axis).
• From $x = 4$ to $10$, there is a rectangle (base $6$, height $4$):

$\text{Area} = \text{base} \times \text{height} = 6 \times 4 = 24$

The total area is:

$8 + 24 = 32$

Answer: $32$

(c) $\int_{10}^{14} f(x) \, dx$

From $x = 10$ to $14$, the graph is below the x-axis. It forms a triangle with:

• Base $4$ and height $-4$ (below the axis).

$\text{Area} = \frac{1}{2} \times 4 \times 4 = 8$

Because the area is below the x-axis, the integral is negative:

$\int_{10}^{14} f(x) \, dx = -8$

Answer: $-8$

(d) $\int_6^{14} f(x) \, dx$

From $x = 6$ to $14$, we split the area into two regions:

1. $x = 6$ to $10$: A rectangle (base $4$, height $4$):

$\text{Area} = 4 \times 4 = 16$

2. $x = 10$ to $14$: A triangle (below the x-axis, area $-8$).

Combining both:

$\int_6^{14} f(x) \, dx = 16 + (-8) = 8$

Answer: $8$

(e) $\int_6^{14} |f(x)| \, dx$

Here we consider the absolute values of the areas:

1. $x = 6$ to $10$: The area is $16$ (already positive).
2. $x = 10$ to $14$: The triangle has area $8$ (take absolute value).

$\int_6^{14} |f(x)| \, dx = 16 + 8 = 24$

Answer: $24$

(f) $\int_4^0 f(x) \, dx$

The integral from $x = 4$ to $0$ is the negative of $\int_0^4 f(x) \, dx$:

$\int_4^0 f(x) \, dx = - \int_0^4 f(x) \, dx = -8$

Answer: $-8$

Final Answers:

(a) $8$
(b) $32$
(c) $-8$
(d) $8$
(e) $24$
(f) $-8$

Let me know if you need further clarification!

Related Questions:

1. How do you calculate definite integrals from geometric shapes?
2. What happens to the sign of an integral when the curve lies below the x-axis?
3. How do absolute value integrals affect the total area calculation?
4. How do you split regions for piecewise functions when evaluating integrals?
5. What is the role of symmetry in calculating areas under a curve?

Tip: Always split the interval into smaller regions where the function remains entirely above or below the x-axis for easier calculation.