Left-Hand Sum
We have formulas to find areas of shapes like rectangles, triangles, and circles (pi, anyone?). What if we want to find the area of a less-reasonable shape? Think of sea monkeys. Sure, we all want...
Right-Hand Sum
There is much debate about who is more awesome...right-handers or south-paws. Would you want Shoeless Joe Jackson on your team, or Nomar Garciaparra? That's why it's better to have a switch hitter...
Comparing Right- and Left-Hand Sums
Left Hand Sums and Right Hand Sums give us different approximations of the area under of a curve. If one sum gives us an overestimate and the other an underestimate,then we can hone in on what the...
Midpoint Sum
We're driving along from right coast to the left coast, and now it's time to take a rest stop at the midpoint sum. Grab some snacks before continuing on. We recommend all flavors of Sun Chips and w...
Trapezoid Sum
All of these summations are starting to feel like Rube Goldberg Machines. Granted, Rube Goldberg Machines are awesome, but do we seriously need this many methods to sum up intervals? Trust us, they...
Definite Integrals of Non-Negative Functions
When f is a non-negative function and a
Definite Integrals of Real-Valued Functions
When we're integrating a non-negative function from a to b, the integral
Properties of Definite Integrals
There are a lot of useful rules for how to combine integrals, combine integrands, and play with the limits of integration. For some functions there are shortcuts to integration.For this whole secti...
Averages with Numbers
To find the average of a bunch of numbers, we add all the numbers and divide by how many there are. Sample ProblemFive students took a test. Their scores were 60, 70, 80, 85, and 87. The aver...
Averages with Functions
Sample Problem
Let f(x) be the function graphed below. We can see that Find a constant m such thatIn other words, find a constant m so that we have this:Answer. . In order to have this eq...