use 5 rectangles to find a lower and upper estimate for the area under a graph from x=0 to x=10???

Question

there is a given graph that has a curve going from (0,1) to (10,7). the curve is concave down, like half a smile :P i dont know how else to explain it. im not sure if this helps thank u

Answer 1

You mean a parabola (the function is something like -x^2)? Alright, let me guide you through this

1. Divide the total x distance by the number of divisions

In this case it will be 10/5 = 2

2. Divide the shape into the amount rectangles desired

In this case it will be 5 rectangles each with a base of 2 units

3. Create a formula to find the total area under the curve

The area of a rectangle is width * length. However, we do not have values in terms of "width" and "length." Therefore we can substitute the base of each rectangle that was discovered in step 2 and the length with the function (f(x)).

4. Solve

2f(2) + 2f(4) + 2f(6) + 2 f(8) + 2f(10)
or
2(f(2)+f(4)+f(6)+f(8)+f(10))






In case you don't know what I mean by the function and f(x), I mean to plug the x value that either the left side or right side of the rectangle is (if the rectangle is on (2,0) and (4,0) then x is going to be 2 or 4 (depending on your choice or your teacher's)) into the function (in this case the function will be something like -x^2).

If you don't have a function, but you have the actual graph, then the f(x) will be the y value at the certain x point.


Hope this helped,
good luck

Answer 2

Draw 5 rectangles under the curve. The base of one should go from 0 to 2 on the x-axis, another from 2-4, and 4-6, 6-8, and 8-10. The left upper corner of these rectangles should not go above the line to find the lower limit. The right upper corner of the rectangle should not go below the line to find the upper limit.

Good luck.