To calculate the definite and indefinite integral of the given function using geometric methods, we first analyze the function, break it into parts, and calculate areas based on geometric shapes.

Given Function

$f(x) = 3\sqrt{5 + 4x - x^2} - 2|3 - x|$ with integration limits $-1$ to $5$.

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Step 1: Analyze Each Term

Term 1: $3\sqrt{5 + 4x - x^2}$

This term represents a semicircle. To confirm, rewrite the expression: $5 + 4x - x^2 = -(x^2 - 4x - 5) = -((x - 2)^2 - 9).$ This simplifies to: $5 + 4x - x^2 = 9 - (x - 2)^2.$ Thus, $\sqrt{5 + 4x - x^2} = \sqrt{9 - (x - 2)^2}$, which is a semicircle with:

• Center at $x = 2$,
• Radius $r = 3$,
• Defined only for $x \in [-1, 5]$ since the radius limits the domain.

The area of a semicircle is: $\text{Area} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (3^2) = \frac{9\pi}{2}.$

Since the semicircle is scaled by $3$, its area becomes: $3 \cdot \frac{9\pi}{2} = \frac{27\pi}{2}.$

Term 2: $-2|3 - x|$

This term is a V-shaped linear function centered at $x = 3$. To calculate the area geometrically:

1. Break it into two regions:
   • For $x < 3$, $|3 - x| = 3 - x$.
   • For $x > 3$, $|3 - x| = x - 3$.
2. Find the slopes and intercepts of the lines:
   • Line 1 ($x < 3$): $-2(3 - x) = -6 + 2x$.
   • Line 2 ($x > 3$): $-2(x - 3) = -2x + 6$.

The resulting "V" shape has two triangles:

• Left triangle ($-1 \leq x \leq 3$): Base = $3 - (-1) = 4$, Height = $6$, Area = $\frac{1}{2} \cdot 4 \cdot 6 = 12$.
• Right triangle ($3 \leq x \leq 5$): Base = $5 - 3 = 2$, Height = $6$, Area = $\frac{1}{2} \cdot 2 \cdot 6 = 6$.

Since the function is scaled by $-2$, the total area is: $-2 \cdot (12 + 6) = -36.$

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Step 2: Combine Areas

The definite integral is the total area of both terms: $\int_{-1}^{5} \big(3\sqrt{5 + 4x - x^2} - 2|3 - x|\big) dx = \frac{27\pi}{2} - 36.$

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Final Answer

$\boxed{\frac{27\pi}{2} - 36}$

Let me know if you want more details on specific steps or further clarifications!

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Related Questions:

1. What is the general method to rewrite quadratic expressions into standard circle form?
2. How can you determine if a function represents a semicircle?
3. What is the relationship between definite integrals and geometric areas?
4. How do absolute values affect the graph of a function?
5. How can scaling constants outside a function affect the resulting area?

Tip: When dealing with definite integrals geometrically, visualize the shapes involved to simplify the process.