Lets take a trip back in timeto geometry. Can you find the area of the following? If so, why?
Now, lets take a trip back to Advanced Algebra. Can you find the area of the region bounded by the line x=0, y=0 , y = 4 and y = 2x+3? If so, how?304
When we find the area under a curve by adding rectangles, the answer is called a Riemann Sum.subintervalpartitionThe width of a rectangle is called a subinterval.The entire interval is called the partition.Subintervals do not all have to be the same size.
If we use subintervals of equal length, then the length of a subinterval is:The definite integral is then given by:
Leibnitz introduced a simpler notation for the definite integral:Note that the very small change in x becomes dx.
IntegrationSymbollower limit of integration upper limit of integration integrandvariable of integration(dummy variable)
We have the notation for integration, but we still need to learn how to evaluate the integral.This will be another day. We will master the Riemann Sum work first! Onwards!!!
After 4 seconds, the object has gone 12 feet. Why?Consider an object moving at a constant rate of 3 ft/sec.Since rate . time = distance:If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.d ?
If the velocity is not constant,we might guess that the distance traveled is still equalto the area under the curve.(The units work out.)Example:We could estimate the area under the curve by drawing rectangles touching at their left corners.We call this the Left-hand Rectangular Approx. Method (LRAM).
We could also use a Right-hand Rectangular Approximation Method(RRAM).
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.
width of subintervalWith 8 subintervals:
Inscribed rectangles are all below the curve:What Riemann Method? Over or under estimate? Concave up or down?What Riemann Method? Over or under estimate? Concave up or down?
Riemann Sums Exercise Handout