This app can be used to find and illustrate approximate values for any definite integral.[br][br]The definite integral of a function over an interval [a, b] is the net signed area between the x-axis and the graph of the function over the interval. When a < b areas above the x-axis contribute positively to the integral and areas below the x-axis contribute negatively to the integral. When b < a area below the x-axis contribute positively to the integral and areas above the x-axis contribute negatively to the integral. Therefore, the in the app above the value of the integral is a number equal to the green area minus the red area.[br][br]In the App[br] Enter the formula for the integrand function f(x) in the input box.[br] Adjust the values of the limits of integration, a and b, via the sliders or input boxes. [br] An approximate value of the definite integral is displayed.
The integral is the net signed area. In the illustration this is the green area - the red area. A positive values indicates that there is more green area than red area, and a negative value indicates that there is more red area than green area.[br][br]One physical appplication of this is if the function f is a velocity function for motion along a line and the input variable x is time, then the integral is the next change in position. A negative value indicates that the object ended up in the negative direction from its starting point, and a positive value indicates that the object ended up in the positive direction from its starting point.
The net area is the positive difference in the green and red areas. It is found by first integrating and then taking an absolute value.[br][br]If the function f is a velocity function, then the net area on the graph is the net distance traveled. I.e. it measures how far from the staring point the object ended (without regard to direction).
The total area is the green area plus the red area. It is found by first taking an absolute value of the function and then integrating. [br][br]If the function f is a velocity function, then the total area on the graph is the total distance traveled, e.g. the amount of change in the odometer on a car.
Check on the Antiderivative box to see the formula for the family of antiderivatives. The value of [math]c_2[/math] is an arbitrary constant. Enter a value for this constant in the input box. The graph of this specific antiderivative function is graphed in dark green.