First Fundamental Theorem of Calculus 2
- Dr. Jack L. Jackson II
Visualizing the Definite Integral on the Graphs of the Integrand and Its Antiderivative
In the App Enter the formula for the integrand function f in its input box in the left window Enter choices for the limits of integration a and b via the sliders or input boxes. In the Left Window You will see the graph of the integrand function f(x) graphed in red. You will see the definite integral expressed as a net signed area between the graph of f(x) and the x-axis. The integral is equal to the green shaded area minus the red shaded area. The notation for the integral and its approximate value are displayed. In the Right Window You will see the formula for an antiderivative function F(x). This is the antiderivative function generated by GeoGebra plus the constant C. Adjust the value of C via the slider or input box to obtain another antiderivative function. The graph of F(x) is in blue. F is an antiderivative of f and f is the derivative of F: F ' (x) = f(x). You will see the values of F(b) and F(a) and the value of F(b) - F(a) in the box and illustrated on the graph. The integral is equal to F(b) - F(a). You will see the integral illustrated as the signed length (F(b) - F(a)) of the illustrated vertical vector. The First Fundamental Theorem of Calculus as it applies to definite integrals is stated.