Urgent homework help! Intergration-right endpoint estimate?

Question

Feel like helping me out? :)
Use the two rectangles (graph in textbook- shows that the width of each retangle is 0.5) to approximate the area contained between the curve and the x-axis using the right endpoint method.
y= (1/2) x (X)^2 between x=2 and x=3
The answer is
A1= 0.5 x 3.125
A2= 0.5 x 4.5
Ans: A1 + A2= 3.8125 units

For A1 I don't understand how to get the 3.125 (I subbed in '2' into the equation) and got the answer as 2.
What am I doing wrong?

Answer 1

Since you are using the "right endpoint method" you want to estimate the height of the function using the right end of the estimation range.

What does that mean?

You are going between x=2 and x=3. Each rectangle is of width 0.5, thus there are (3-2)/0.5 = 2 rectangle (but you knew that already) A1 and A2.

A1 goes from x=2 to x=2.5
A2 goes from x=2.5 to x=3

since you are using the right endpoint method, you want to estimate the height of the graph in the first inteval by the height at x=2.5 (thus the height of A1 is f(2.5)).

Likewise the height of A2 is f(3).

Therefore the area of A1 is 0.5•f(2.5) = 0.5(1/2(2.5)^2) = 0.5•(1/2•6.25) = 0.5 • 3.125
and the area of A2 is 0.5•f(3) = 0.5(1/2(3)^2 = 0.5•(1/2•9) = 0.5 • 4.5

And the estimated area under the curve is area of A1 + area of A2 = 0.5 • 3.125 + 0.5 • 4.5 = 0.5(3.125+4.5) = 0.5 • 7.625 = 3.8125

Answer 2

Well i am not sure how it exactly works but.... the hint is to use the RIGHT ENDPOINT ESTIMATE

For A1 the rectangle start point is from 2 and ends at 2.5
For A2 the rectangle start point is from 2.5 and ends at 3

Hence the right end for A1 = 2.5
A2 = 3
Substituting these values gives you the answer as shown

Substituting these values gives you the answers

Answer 3

send me the graph.I dont understand whats going on?