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Definite Integrals
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Calculus
Definite Integrals
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Definite Integrals Exercises
LeftHand 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 lessreasonable shape? Think of sea monkeys. Sure, we all want...
RightHand Sum
There is much debate about who is more awesome...righthanders or southpaws. 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 LeftHand Sums
Sample ProblemLet f be an increasing function on [a,b] and let R be the region between the graph of f and the xaxis on [a,b].Will LHS(n) be an over or underestimate of the area of R?Will RHS(n)...
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 NonNegative Functions
When f is a nonnegative function and a
Definite Integrals of RealValued Functions
When we're integrating a nonnegative function from a to b where a
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, you 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 average...
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 e...
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