7.1 Measurement and the Definite Integral; Arc Length
Integrating with respect to x
This first applet is for use when integrating with respect to x. The applet allows you to view a Riemann Sum approximation for the integral as well. Notice that when integrating with respect to x, the Riemann Sum has vertically oriented rectangles. Likewise, the functions should be expressed in terms of y=f(x). Below, "f(x)" refers to the function that defines the border along the top side of the shaded region (shown in red), whereas, "g(x)" refers to the function that defines the border along the bottom side of the shaded region (shown in blue).
Arc Length of a Curve
The applet below shows the approximation for arc length by using straight line segments. Move sliders a and b to change the position of the end points of the interval, A and B, respectively. The slider for n changes the number of segments used to estimate the arc legth of the curve. When n is large more points are used and, therefore, the line segments are very small. The sum of the lengths of all line segments gives an estimate for the arc length of the curve. In addition, the exact value of the arc length found by integration is shown as well. You can also change the function f(x) by entering the function in the input line at the bottom. For example, to estimate the arc length of x^3 from x=-1 to x=1, type "f(x)=x^3" in the input bar and then adjust the slider for a=-1 and b=1. Note: tapping on the value of n will allow you to increase the number of line segments one unit at a time.