© 2015 Shmoop University, Inc. All rights reserved.
Definite Integrals

Definite Integrals

Definite Integrals Exercises

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

Sample ProblemLet f be an increasing function on [a,b] and let R be the region between the graph of f and the x-axis on [a,b].Will LHS(n) be an over- or under-estimate 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 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 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...
Noodle's College Search