https://file.aiquickdraw.com/mathbot/file/1725928491166w841kw1k.jpg

To solve the given integral using the area interpretation, we need to analyze the function geometrically.

The integral provided is:

\[
\int_0^2 \left( |6x - 2| + \sqrt{4 - x^2} \right) \, dx
\]

This integral has two components: 
1. \( |6x - 2| \)
2. \( \sqrt{4 - x^2} \)

### Step 1: Analyze \( |6x - 2| \)

The absolute value function \( |6x - 2| \) can be broken into two cases:

- For \( x \geq \frac{1}{3} \), \( 6x - 2 \geq 0 \), so \( |6x - 2| = 6x - 2 \).
- For \( x < \frac{1}{3} \), \( 6x - 2 < 0 \), so \( |6x - 2| = -(6x - 2) = 2 - 6x \).

Thus, the integral can be split into two parts:
\[
\int_0^{\frac{1}{3}} (2 - 6x) \, dx + \int_{\frac{1}{3}}^2 (6x - 2) \, dx
\]

These integrals are straightforward calculations of linear functions.

### Step 2: Analyze \( \sqrt{4 - x^2} \)

The function \( \sqrt{4 - x^2} \) represents the upper half of a circle with radius 2 centered at the origin. The area under this curve from \( x = -2 \) to \( x = 2 \) is the area of a semicircle with radius 2.

The formula for the area of a semicircle is \( \frac{1}{2} \pi r^2 \). Since the integral bounds are from 0 to 2, we are looking at one-quarter of the circle, so the area will be:

\[
\frac{1}{4} \pi (2^2) = \pi
\]

### Step 3: Combine the results

We now compute the two components:

#### Integral of \( |6x - 2| \)

1. From \( x = 0 \) to \( x = \frac{1}{3} \):

\[
\int_0^{\frac{1}{3}} (2 - 6x) \, dx = \left[ 2x - 3x^2 \right]_0^{\frac{1}{3}} = 2 \times \frac{1}{3} - 3 \times \left( \frac{1}{3} \right)^2 = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}
\]

2. From \( x = \frac{1}{3} \) to \( x = 2 \):

\[
\int_{\frac{1}{3}}^2 (6x - 2) \, dx = \left[ 3x^2 - 2x \right]_{\frac{1}{3}}^2 = \left( 3(2^2) - 2(2) \right) - \left( 3\left( \frac{1}{3} \right)^2 - 2 \times \frac{1}{3} \right)
\]

\[
= (12 - 4) - \left( 3 \times \frac{1}{9} - \frac{2}{3} \right) = 8 - \left( \frac{1}{3} - \frac{2}{3} \right) = 8 + \frac{1}{3} = \frac{25}{3}
\]

Thus, the total contribution from \( |6x - 2| \) is:

\[
\frac{1}{3} + \frac{25}{3} = \frac{26}{3}
\]

#### Integral of \( \sqrt{4 - x^2} \)

As discussed, this integral gives \( \pi \).

### Final Answer

The total area under the curve is the sum of the two parts:

\[
\frac{26}{3} + \pi
\]

Thus, the exact value of the definite integral is:

\[
\boxed{\frac{26}{3} + \pi}
\]

---

Would you like further clarification on any of these steps? Here are some related questions you might find interesting:

1. How do we calculate the area under a general absolute value function?
2. Why does the function \( \sqrt{4 - x^2} \) represent a semicircle?
3. How does the area interpretation differ from traditional integration techniques?
4. Can we compute this integral using substitution instead of area interpretation?
5. What are some other cases where absolute value functions arise in integration?

**Tip:** When dealing with absolute value functions, it's useful to break the function into intervals where the expression inside the absolute value is positive or negative.

Use area interpretation to find the exact value of the definite integral ∫₀² (|6x - 2| + √(4 - x²)) dx.

This detailed solution covers the use of area interpretation to evaluate the definite integral ∫₀² (|6x - 2| + √(4 - x²)) dx. It includes the geometry of absolute value functions and semicircles and is suitable for advanced high school or introductory college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation