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The Second Fundamental Theorem of Calculus allows us to calculate definite integrals without Riemann sums. This gives us the area bounded by the curve of f f f, the x-axis, and the lines x = a x = a x = a and x = b x = b x = b. The above statement means that we can evaluate the definite integral of a function on by taking the difference between the indefinite integral of the function evaluated at a a a and the indefinite integral of the function evaluated at b b b. Let the function F ( x ) F(x) F ( x ) be defined byį ’ ( x ) = d d x ∫ 3 x ( 6 t 4 + sin ⁡ ( t ) ) d t = 6 x 4 + sin ⁡ ( x ) F’(x) = \frac f(x)\,dx = F(x)\Big|_a^b = F(b) - F(a) ∫ a b ​ f ( x ) d x = F ( x ) ∣ ∣ ​ a b ​ = F ( b ) − F ( a ) Let f f f be a continuous function on the interval. We also refer to it as the Fundamental Theorem of Differential Calculus. The First Fundamental Theorem of Calculus shows that integration and differentiation are inverse operations. In 1823, French mathematician Austin-Louis Cauchy rigorously proved the theorem. In the later 17th century, English mathematicians Isaac Barrow and Isaac Newton and German mathematician Gottfried Wilhelm Leibniz separately developed the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus was first articulated in 1668 by Scottish mathematician James Gregory.

The Fundamental Theorem of Calculus became the connective tissue that linked these essential operations. Integration and differentiation were initially developed separately because we did not suspect that they were related. Hannah Fry shares what this theorem actually means, how it’s calculated, and what it can further allow us to do. You might also hear this theorem referred to as the “FTC.” The First Fundamental Theorem of Calculus reveals that integration is the inverse process of differentiation, while the Second Fundamental Theorem of Calculus illuminates the relationship between the integral and the antiderivative function. The Second Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus The Fundamental Theorem of Calculus helps to define the relationship between integration and differentiation. What Is the Second Fundamental Theorem of Calculus?Ħ Example Exercises of Fundamental Theorem of Calculus What Is the First Fundamental Theorem of Calculus?

What Is the Fundamental Theorem of Calculus?