The images you provided show the graph of a piecewise linear function and three definite integrals to evaluate using the graph. Let's analyze the graph and solve each integral by interpreting the areas under the curve.

Analysis of the Graph:

The graph consists of straight-line segments and a semi-circular arc. The key points where the line changes or the arc begins/ends seem to be approximately at the following coordinates:

• The line starts at approximately $(-2, 0)$.
• The slope changes at $(0, -2)$, forming a downward slope to the x-axis.
• There is a semi-circle centered around the x-axis, between $(2, -3)$ and $(4, -3)$.
• The final linear segment rises from the endpoint of the arc.

Integrals to Evaluate:

We are tasked with evaluating these integrals:

1. $\int_{-2}^{2} f(x) \, dx$
2. $\int_{2}^{4} f(x) \, dx$
3. $\int_{4}^{8} f(x) \, dx$

We will interpret these integrals using the area under the curve.

1. $\int_{-2}^{2} f(x) \, dx$

• This integral covers the triangular area from $(-2, 0)$ to $(0, -2)$, and another triangular area from $(0, -2)$ to $(2, 0)$.
• The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$.
• Each triangle has a base of 2 units and a height of 2 units, resulting in an area of $2 \text{ units}^2$.
• Both triangles are below the x-axis, so their areas are negative.
• Therefore, the integral value is $-2 - 2 = -4$.

2. $\int_{2}^{4} f(x) \, dx$

• This integral corresponds to the area of the semi-circle with a radius of 1 unit below the x-axis.
• The area of a semi-circle is $\frac{1}{2} \pi r^2$.
• Substituting $r = 1$, the area becomes $\frac{1}{2} \pi (1)^2 = \frac{\pi}{2}$.
• Since the semi-circle is below the x-axis, the integral value is $-\frac{\pi}{2}$.

3. $\int_{4}^{8} f(x) \, dx$

• This integral involves a triangular area from $(4, 0)$ to $(8, 3)$.
• The base of this triangle is 4 units, and the height is 3 units.
• The area of the triangle is $\frac{1}{2} \times 4 \times 3 = 6$.
• The triangle is above the x-axis, so its area is positive.
• Therefore, the integral value is $6$.

Summary of Integral Values

1. $\int_{-2}^{2} f(x) \, dx = -4$
2. $\int_{2}^{4} f(x) \, dx = -\frac{\pi}{2}$
3. $\int_{4}^{8} f(x) \, dx = 6$

Would you like more details on these calculations or have any questions about interpreting the graph?

Related Questions:

1. How do you calculate the area of a triangle when its vertices are not aligned with the x-axis or y-axis?
2. What is the significance of a negative integral value in terms of graph interpretation?
3. How can the area under a curve be approximated when the shape is not standard, like a semi-circle?
4. What techniques are used to find the integral of functions with both linear and curved sections?
5. How does the definite integral change if the limits of integration are reversed?

Tip:

When interpreting definite integrals as areas, remember that areas above the x-axis contribute positively to the integral, while areas below the x-axis contribute negatively.