Essential Knowledge 3.2B2

Essential Knowledge 3.2B2 Students will know that definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.

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Now it's time to show you situations where calculus becomes essential, because areas are no longer simple shapes

Example

A car is rolling along the road at 5 ft/sec. Then the car begins accelerating such that the velocity is given by the equation:

v = 5t² + 5.

Determine the distance the car will have traveled 3 seconds after it begins accelerating.

a) Use a left Riemann sum with 4 equal subintervals to approximate the distance traveled.

b) Use a right Riemann sum with 6 equal subintervals to approximate the distance traveled.

c) Use a midpoint Riemann sum with 3 equal subintervals to approximate the distance traveled.

d) Use a trapezoidal Riemann sum with 3 equal subintervals to approximate the distance traveled.

Graphically

First we need the graph. I'm going to upload an applet

I'm very thankful to whoever worked so hard on this because it's pretty great. We can use this to show exactly what an integral is, and how limits allow us to make our answer mathematically exact.

Anyways, the first thing to notice is that we have no geometric shapes that look like this. Therefore there is no area formula to find the area under this curve. So how will we do this?

Mathematicians had a brilliant idea. We can just make a bunch of rectangles under this curve, and in that way get a pretty good estimate of the area under it. Let's experiment with this to see what I mean.

First, go to the "rectangles" box and type in 2, then press enter on your keyboard. Now to go the button "left endpoints" and click it. You'll see two rectangles under the curve.

As you can see, two rectangles is a pretty poor estimate for the area under the curve. But let's kick it up to 6 rectangles. Now you can see the estimate is getting much better. Try 30 rectangles. Hey now that's almost the same thing as the area. Now try 200 rectangles

Wow. That looks like exactly the area under the curve. Do you see where we're going with this? To make it perfect, we would need infinity rectangles, each with an infinitely small width. That's exactly how an integral was created.

But anyways, here we're just going to estimate with a finite number of rectangles. Later we'll do the infinite rectangles.

a) Use a left Riemann sum with 4 equal subintervals to approximate the distance traveled.

Okay, so we want 4 rectangles whose top left corner touches the curve. At any time you can input this into the gadget to see how this should look.

So, let's think about this. If the interval from 0 to 3 seconds must be divided into 4, we get that each interval must be

(3-0)/4 = 0.75 seconds wide

How tall will each rectangle be? Well, let's see what the velocity is at those different times using our equation v = 5t² + 5.

First rectangle

let t = 0

v = 5(0)² + 5 = 5 ft/sec

Second rectangle

let t = 0.75

v = 5(0.75)² + 5 = 7.8125 ft/sec

Third Rectangle

let t = 1.5

v = 5(1.5)² + 5 = 16.25 ft/sec

Fourth Rectangle

let t = 2.25

v = 5(1.5)² + 5 = 30.3125 ft/sec

To find the areas of the rectangles, simply multiply the base (all rectangles are 0.75 sec wide, so this number will be the same each time) times the height (velocity)

1st rectangle

A = bh = (0.75 sec)(5 ft/sec) = 3.75 ft

2nd rectangle

A = bh = (0.75 sec)(7.8125 ft/sec) = 5.859375 ft

3rd rectangle

A = bh = (0.75 sec)(16.25 ft/sec) = 12.1875 ft

4th rectangle

A = bh = (0.75 sec)(30.3125 ft/sec) = 22.734375 ft

Finally, we can add these 4 rectangles together and we get 44.53125 ft. As you can see this agrees with the applet (except they rounded a bit)

b) You can do this one

c) You can do this one too, just find the velocity at the center of the intervals

d) OK I'll do this one. JK. You can do this. Just make trapezoids instead of rectangles. And remember that the equation for a trapezoid is b*(1/2)(h1 + h2). h1 and h2 are the velocities at the beginning and the end of your intervals.