"A train moves along a track at a steady rate of 75 miles per hour from 8AM to 9AM. What is the total distance traveled by the train?"
We know the formula "Distance= Rate x Time", so we find that the answer is 150 miles. It might be this simple, but, when you look at this on the graph, you can see that the distance traveled by the train is the AREA of the rectangle whose base is the time interval [7,9] and the height at each point is the value of the constant velocity function v=75.
With these kinds of problems you can use Midpoint Rectangular Approximation Method, or left-hand RAM or right-hand RAM.
In the picture below you will be able to see how we can find the approximate area under the curve y=x^2 from x=0 to x=3.
We know the formula "Distance= Rate x Time", so we find that the answer is 150 miles. It might be this simple, but, when you look at this on the graph, you can see that the distance traveled by the train is the AREA of the rectangle whose base is the time interval [7,9] and the height at each point is the value of the constant velocity function v=75.
With these kinds of problems you can use Midpoint Rectangular Approximation Method, or left-hand RAM or right-hand RAM.
In the picture below you will be able to see how we can find the approximate area under the curve y=x^2 from x=0 to x=3.
With the RAM approximation method, we have to add the products of the length of the interval of the rectangle and the height of it. We can find the height of the rectangle by inserting the x-value of where it hits the curve.
Here is a video that explains this concept in depth.
Here is another video explaining this same concept.
Another example using the midpoint rule
Midpoint Rule
In the midpoint rule, the rectangles height is determined by where the midpoint of
the box hits the curve, and the rectangles are draw on curve as shown. We are
splitting the area up into four regions. Once again the width of each rectangle is
0.25. In the picture the midpoint of the box determines the height, so we need to
know the half-way point between each interval. To find this, for the first interval,
we do (0 + 0.25)/2 = 0.125. Then we do (0.25 + 0. 5)/2 = 0.375, and so forth. You
will get the x values below the figure. Then we take each of these midpoint values
and put them into our equation, . The corresponding y values are shown 2
in the figure. Now we want to find the area of each rectangle and add it together.
I will use the formula A = LW. In our case, the width of each box is 0.25.
A = 0.9844(0.25) + 0.8594(0.25) + 0.6094(0.25) + 0.2344(0.25) = 0.6719.
Here is a video that explains this concept in depth.
Here is another video explaining this same concept.
Another example using the midpoint rule
Midpoint Rule
In the midpoint rule, the rectangles height is determined by where the midpoint of
the box hits the curve, and the rectangles are draw on curve as shown. We are
splitting the area up into four regions. Once again the width of each rectangle is
0.25. In the picture the midpoint of the box determines the height, so we need to
know the half-way point between each interval. To find this, for the first interval,
we do (0 + 0.25)/2 = 0.125. Then we do (0.25 + 0. 5)/2 = 0.375, and so forth. You
will get the x values below the figure. Then we take each of these midpoint values
and put them into our equation, . The corresponding y values are shown 2
in the figure. Now we want to find the area of each rectangle and add it together.
I will use the formula A = LW. In our case, the width of each box is 0.25.
A = 0.9844(0.25) + 0.8594(0.25) + 0.6094(0.25) + 0.2344(0.25) = 0.6719.
Overall, this chapter seems to be a completely different concept from what we have been learning so it's cool to feel like we have a fresh start. This section seems to be more logical.
Also, Happy Holidays from Darrin and I!