Integration of a Parabolic Region

Overview

A concise walkthrough of solving the quadratic equation $16 - x^2 = 0$ to find the intersection points, followed by evaluating the definite integral $\displaystyle \int_{-4}^{4} (16 - x^2)\\,dx$ to obtain the exact area under the parabola. The key takeaway is the step‑by‑step antiderivative evaluation leading to the exact area $\frac{256}{3}\approx 85.33\ ext{ cm}^2$.

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Phases

#	Phase Name	Duration	Description
1	Intro	~1s	Header "Task 2" fades in at the upper‑left corner.
2	Solve for $x$	~2s	The equation "16 - x^2 = 0" is written, then the solutions $x_1 = 4,\\; x_2 = -4$ appear.
3	Set up Integral	~1s	The integral expression $y = \int_{-4}^{4} (16 - x^2)\\,dx$ is displayed.
4	Antiderivative Step	~1.5s	The antiderivative $[16x - \frac{x^3}{3}]_{-4}^{4}$ is written.
5	Substitute Limits	~1.5s	The evaluated limits are shown as a two‑line expression.
6	Simplify Result	~1s	The simplification to $\frac{256}{3}$ is written.
7	Exact Area Statement	~1s	Final statement "exact area = \frac{256}{3} \approx 85.33 \ ext{ cm}^2" appears.
8	Pause / End	~2s	Brief pause before the scene ends.

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Layout

┌─────────────────────────────────────────────┐
│ Header (Task 2)                              │
│                                             │
│              MAIN (equations)               │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional, omitted here)            │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Header	Small text "Task 2" in the upper‑left corner.	Appears once at start
Main	Sequence of MathTex objects (part a, integral set‑up, antiderivative, limit substitution, simplification, exact area). Arranged vertically, left‑aligned.	Primary visual focus
Caption	None (no persistent footer needed).	—

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Notes

• Assumptions: No additional visual elements (graphs, axes) are required beyond the algebraic steps provided. The total runtime is kept under 10 seconds for a clear, brisk presentation.
• All equations are displayed using MathTex; scaling is chosen to keep readability while fitting the vertical stack within the main area.
• A short pause of 0.5 s after the header and a final 2 s pause at the end give viewers time to absorb the result.
• No extra scenes are needed; the entire animation fits within a single Scene class.
• No caption/footer is used because the step titles are already embedded in the equations.