# Area Under a Curve - Right points

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? • ...using four rectangles... – Anton Vrdoljak Jul 31 '20 at 7:14
• Is that not 4 rectangles? – Eight 8 Jul 31 '20 at 7:15
• Your figure showing 5 rectangles.... – Anton Vrdoljak Jul 31 '20 at 7:17

## 2 Answers

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 $$f(x_i)\cdot \Delta x$$.

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.

• Thanks so much! However, I'm still a little confused. Is it possible for you to write how you would solve the problem? :) – Eight 8 Jul 31 '20 at 7:26
• @Eight8 A better idea would be that you try again, now that you know what you did wrong. You can edit your question with your second attempt, and if you then drop a comment, I will be happy to re-check your solution. Also, for formatting of your question, I suggest you check out this link: math.meta.stackexchange.com/questions/5020/… – 5xum Jul 31 '20 at 7:31
• I'm still struggling to get the answer, any recommendations? I posted my 2nd attempt. – Eight 8 Jul 31 '20 at 8:46
• @Eight8 I edited my answer. – 5xum Jul 31 '20 at 8:54
• Thank you so much!!!!! – Eight 8 Jul 31 '20 at 9:11

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

• Thanks! Here I only see 3 rectangles and I need 4? – Eight 8 Jul 31 '20 at 7:40
• Ok, I used right endpoint rule method with 4 intervals, usually it will result with 4 rectangles, but we have a special case, i.e. $f(x)=1 - x^2$, so try by yourself my approach, but with 5 intervals! In other words $P_1$ will have coordinates $(0.2, 0)$... – Anton Vrdoljak Jul 31 '20 at 7:44
• If I may ask, which website are you using? I would like to generate a graph like this. Thanks. – Eight 8 Jul 31 '20 at 7:47
• I did this graph in GeoGebra... – Anton Vrdoljak Jul 31 '20 at 7:51