First Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus states that a definite integral over a closed interval is equal to the di...
The First Fundamental Theorem of Calculus states that a definite integral over a closed interval is equal to the difference between the area of the region bounded by the curve and the x-axis and the area of a rectangle with the same base and height as the region.
Formally:
where:
(\int_a^b f(x) dx) is the definite integral from (a) to (b)
A is the area of the region bounded by the curve and the x-axis
B is the area of the rectangle with the same base and height as the region
Examples:
Area of a triangle: If the curve is a triangle with height (h) and base (b), the area is (\frac{1}{2} bh).
Area of a semicircle: If the curve is a half circle with radius (r), the area is (\frac{1}{2} \pi r^2).
Intuitively:
Imagine dividing the region under the curve into thin strips. The total area of these strips is approximated by the sum of the areas of the individual strips. The theorem tells us that the total area is equal to the limit of the sum of the areas of the strips as the width of the strips approaches 0.
Applications:
The First Fundamental Theorem of Calculus has numerous applications in different fields, including:
Calculating areas and volumes of various shapes
Finding the definite integral of a function
Establishing the connection between definite and indefinite integrals
The theorem is a fundamental tool in calculus and is used extensively in numerous advanced concepts, including the Second Fundamental Theorem of Calculus and the definite integral definition of a derivative