https://file.aiquickdraw.com/mathbot/file/1733091711374x1a1r2us.jpg

To compute the integral \(\int_{-1}^{9} f(x) \, dx\), we will analyze the area under the curve given in the graph. Here's the step-by-step solution:

### Step 1: Break Down the Graph
The graph consists of:
1. A triangle from \(x = -1\) to \(x = 1\).
2. A semicircle from \(x = 1\) to \(x = 9\).

The integral can be computed as the sum of these two areas:
\[
\int_{-1}^{9} f(x) \, dx = \text{Area of Triangle} + \text{Area of Semicircle}.
\]

### Step 2: Compute the Area of the Triangle
The triangle's vertices are at:
- \((-1, 0)\),
- \( (1, 1)\),
- \( (1, 0)\).

The base of the triangle is from \(-1\) to \(1\), so its length is \(2\), and the height is \(1\). The area of a triangle is:
\[
\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 1 = 1.
\]

### Step 3: Compute the Area of the Semicircle
The semicircle has:
- Radius \(r = 4\) (since the diameter is from \(x = 1\) to \(x = 9\)).
- The area of a semicircle is half the area of a circle:
\[
\text{Area of Semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4)^2 = 8\pi.
\]

### Step 4: Combine the Results
Adding the two areas:
\[
\int_{-1}^{9} f(x) \, dx = \text{Area of Triangle} + \text{Area of Semicircle} = 1 + 8\pi.
\]

### Final Answer:
\[
\int_{-1}^{9} f(x) \, dx = 1 + 8\pi.
\]

Let me know if you need more explanation or related help! Here are some follow-up questions to expand understanding:

1. How do you compute the area of a quarter-circle?
2. What happens if the semicircle is below the \(x\)-axis?
3. How would you compute the integral if the triangle slanted downward?
4. Can you apply symmetry to simplify the calculation of the semicircle's area?
5. What are the units of this integral if the graph represents a physical scenario?

**Tip:** Always sketch a graph or label points to ensure accuracy when working with geometry-based integrals.

Compute the integral ∫ from -1 to 9 of f(x)dx, where the graph of f is the red curve given in the following figure.

This detailed solution demonstrates how to compute the integral ∫ from -1 to 9 of f(x)dx using geometric shapes such as a triangle and a semicircle. Explore the application of the Fundamental Theorem of Calculus with step-by-step guidance.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation