Area Under a Curve - Right points

Question

I'm really conflicted about whether I did this correctly. Feedback would be appreciated!

2nd Attempt:

I'm still really confused because I've tried so many variations of this problem and still haven't gotten the correct methodology (the correct answer is 0.5313 which I got through a calculator) but I can't seem to get it?

Answer 1

Your solution is incorrect. You wrote

$$Area=0\cdot f(0.25) + 0.25 \cdot f(0.25) + 0.5 \cdot f(0.25) + 0.75 \cdot f(0.25)$$

which is equal to $$x_0 \cdot f(\Delta x)+x_1 \cdot f(\Delta x)+x_2 \cdot f(\Delta x)+x_3 \cdot f(\Delta x)$$

which means you

1. Took left endpoints instead of right ones (i.e., $0,0.25, 0.5$ and $0.75$ instead of $0.25, 0.5, 0.75, 1$
2. Calculated the expression $f(\Delta x)\cdot x_i$ instead of $f(x_i)\cdot \Delta x$.

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Your second attempt is better, but still incorrect. Answer the following questions:

1. If you split the interval $[0, 1]$ into $4$ equal-length intervals, what are these intervals?
2. What are the edge points of these intervals?
3. From (2), what, therefore, are the values of $x_i$ in the expression $$\sum_{i=1}^4 f(x_i)\Delta x ?$$
4. How is $\Delta x$ defined? The difference between which two numbers is $\Delta x$?

I suggest you try and re-read either your lecture notes or textbook, and answer the four answers above. I can check your answers if you do, and if you do, I believe you will see what you are doing wrong.

Answer 2

Hint: Approximation of an integral/area using right endpoint rule method requires that the rectangles touch the curve with their top-right corners.

Figure: