Which is the best approximation? (Mid ordinate)?

Question

The diagram to the right shows the region enclosed by the curve y = x^2, the lines x =1 and x = 5 and the x axis.

Find the approximation for this area using mid ordinate with:
1 strip
2 strips
and 4 strips.

Now i've done that, and with each of the strips I came up with the same answer 12.5 So that part is down, but it's the next part I'm a little confused about.

~Which do you think is the best approximation?
Why?

Would it be the first one because it requires less calculations for the same answer or the last one because it is more specific?
Or are the all the same?

Answer 1

If you did the first part right, there is no way the the answers with 1, 2, and 4 strips would come out the same.

Let's try it for two strips:

The total width of the region is 4 so each of the two strips has width 2.

The first strip goes from x = 1 to x= 3, the second goes from x = 3 to x = 5.

So the mid point x values are 2 and 4.

The y values for these would by 2^2 and 4^2 which is 4 and 16. (Recall, y = x^2, which is x times x)

So one strip is 2 by 4, while the other is 2 by 16. The first has an area of 8 while the second has an area of 32. The total computed area using 2 strips is therefore 8 + 32 = 40.

Now you can do it for one strip (i.e. width = 4) and 4 strips (i.e. width = 1) and compare.

As for determining which is the best approximation, I suggest you draw the region, and then the 7 strips. I think you'll understand which is best and why.