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The Fundamental Theorem of Calculus is a cornerstone of calculus, linking differentiation and integration. It comprises two parts: the first provides an antiderivative for the integrand, while the second, known as the Evaluation Theorem, aids in calculating definite integrals. This theorem is crucial for computing areas under curves and has applications in physics, economics, and engineering. Understanding its components and applications is essential for solving complex problems in various fields.

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## Definition and Components

### Antiderivatives

Antiderivatives are functions that are the inverse of derivatives

### Limits of Integration

The limits of integration define the interval over which the area is being calculated

### Continuity and Differentiability

The function must be continuous and have an antiderivative for the theorem to be applied

## Applying the Fundamental Theorem of Calculus

### Finding Antiderivatives

To apply the theorem, one must first identify an antiderivative of the function to be integrated

### Evaluating Definite Integrals

The theorem allows for the straightforward calculation of the area under a curve by using antiderivatives

### Best Practices

Precision in computation and ensuring continuity and existence of an antiderivative are essential for accurate results

## Implications and Applications

### Connection between Differentiation and Integration

The theorem establishes a profound connection between the operations of differentiation and integration

### Real-World Applications

The theorem has practical applications in physics, economics, and engineering, among other disciplines

### Importance in Mathematics

The theorem is a fundamental concept in calculus and has broad implications across mathematics