# Definite Integrals Examples

#### 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 wan...

#### 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 hitte...

#### Error in Left- and Right-Hand Sums

How far off are the left/right hand sums? It's sort of like thinking about how much a four-year-old colors outside of the lines with his spanking new, easy grip Crayola's. If f is monotonic (eithe...

#### 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...

#### 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, th...

#### 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

#### Single-Function Properties

These properties require a little more explanation. We're still assuming f is an integrable function.Let c be a constant. Then As an example, let f = x on [0, b] and let c = 3. When we stretch f...

#### Talking About Two Functions

Now another function is going to join the party. Assume both f and g are integrable functions.The integral of a sum is the sum of the integrals. In symbols,Also, the integral of a difference is t...

#### Thinking Backwards

"Thinking backwards" shows up in a lot of places. That means it must be important! Here, "thinking backwards" means instead of using a known integral value to evaluate an expression, we'll work b...

#### 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...

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