When we find the area under a curve by adding rectangles, the answer is called a Riemann Sum. subinterval partition The 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.
subinterval partition If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better. Why? if P is a partition of the interval
is called the definite integral of over. 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.
Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.
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!!!
time velocity 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 equal to 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). Approx. area:
We could also use a Right-hand Rectangular Approximation Method(RRAM). Approx. area:
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). Approx area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.
Approximate area: width of subinterval With 8 subintervals: The exact answer for this problem is.
Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve: What Riemann Method? Over or under estimate? Concave up or down?