Week+of+12.1.11-12.8.11

Part I- What Can the Area "under" a Curve Tell Us?
In class on 11/17, we looked at how the **area** under a //velocity/time// graph gave us the **distance** that an object (in this case, a "punkin'") traveled. This made sense if we looked at the units on the //axes// of this graph: the //y//-axis was measured in ft/sec and the //x//-axis was measured in seconds. When we found the area, we ended up //multiplying// a vertical value (ft/sec) by a horizontal value (sec). (ft/sec)x(sec)=ft !

In "The Snowball" example (part c.), you were asked what the **area** under that curve represented in terms of the growing snowball, its radius, and its volume. Since the //y//-axis represented the **//rate//** at which the volume of the snowball was changing and the //x//-axis represented the //radius// of the snowball at a given instant, we should look at the units on each axis to help us figure out what the **area** would represent. *This would be easier if I had given the right units to begin with! The units on the //y//-axis, the rate at which volume is changing, should be inches cubed //per// inch (not in/sec). Furthermore, this unit simplifies to inches squared.* This is kind of a weird unit, but we can think about it like this: At any given //x//-value on that graph (say, //x//=2), the //y//-value on the graph tells us the //**rate**// at which the volume of the snowball is increasing //at the instant// the radius of the snowball is 2 inches. So we could say, "When the radius of the snowball is 2 inches, the volume of the snowball is increasing 16pi inches squared." This should make sense because at that instant, we are "adding" one more "layer" of surface area to our sphere, and the surface area of a sphere of radius 2 is 16pi sq.in.

Now, what would the **area** between this curve and the //x//-axis represent? ... Post a response to this question under the "Discussion" tab at the top of this wiki p ag e **BY TUESDAY 12/6**.

===We'll talk more in class about what the **area** under a curve represents. . . but you can practice //finding// (or //approximating//) the area under a curve on your own. . .===


 * Part II- Approximating the Area "under" a Curve **

In the "Punkin' Chunkin'" example, it was easy to find the area because the velocity function was linear and created two right triangles with the //x//-axis. (It is also easy to //find// the area in "The Fundraiser" problem because that was also a linear function.)

In "The Snowball" problem, it was not as easy to exactly find the area because we couldn't draw a nice (known) shape that exactly fit the space under the curve. BEFORE moving on, respond to this question: Snowball Question Now, view this presentation on Intro to Area Under a Curve :  You'll need to click on the tabs at the bottom of the Sketchpad document to advance the presentation. Respond to this question: Area Approximation

Complete the following problems:

//Optional:// For extra help, watch the video on approximating area using Riemann sums: [|Right-Hand Sum Example]