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

enter image description here

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?

enter image description here

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

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.

  • $\begingroup$ Thanks so much! However, I'm still a little confused. Is it possible for you to write how you would solve the problem? :) $\endgroup$ – Eight 8 Jul 31 '20 at 7:26
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    $\begingroup$ @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/… $\endgroup$ – 5xum Jul 31 '20 at 7:31
  • $\begingroup$ I'm still struggling to get the answer, any recommendations? I posted my 2nd attempt. $\endgroup$ – Eight 8 Jul 31 '20 at 8:46
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    $\begingroup$ @Eight8 I edited my answer. $\endgroup$ – 5xum Jul 31 '20 at 8:54
  • $\begingroup$ Thank you so much!!!!! $\endgroup$ – 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.

Figure: enter image description here

  • $\begingroup$ Thanks! Here I only see 3 rectangles and I need 4? $\endgroup$ – Eight 8 Jul 31 '20 at 7:40
  • $\begingroup$ 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 $(0.2, 0)$... $\endgroup$ – Anton Vrdoljak Jul 31 '20 at 7:44
  • $\begingroup$ If I may ask, which website are you using? I would like to generate a graph like this. Thanks. $\endgroup$ – Eight 8 Jul 31 '20 at 7:47
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    $\begingroup$ I did this graph in GeoGebra... $\endgroup$ – Anton Vrdoljak Jul 31 '20 at 7:51

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