We just started this topic in class and I am not quite understanding. I hope someone can guide me through this problem. Estimate the area under the graph of $ƒ(x)=1+x^2$ from $x=−1$ to $x=3$ using four rectangles and right endpoints

  • $\begingroup$ For this problem you actually don't need to find the anti-derivative. This is an estimation problem, where you draw rectangles from the x axis until the point where the right side of the rectangle reaches the line of the function. You then estimate the combined area of the triangles. $\endgroup$
    – PhzksStdnt
    Dec 3, 2014 at 21:55
  • $\begingroup$ Wolfram Alpha is really powerful nowadays. :) $\endgroup$
    – apnorton
    Dec 3, 2014 at 22:53
  • $\begingroup$ @anorton I know but I want to learn how to do this problem and sometimes Wolfram Alpha does not show you $\endgroup$
    – Csci319
    Dec 3, 2014 at 22:56
  • $\begingroup$ @BS319 True; I thought the picture was kind-of nice, though. :) $\endgroup$
    – apnorton
    Dec 3, 2014 at 22:57

1 Answer 1

  1. Partition the interval $[-1,3]$ on the $x$ axis into 4 equal parts (so at $x=0,1,2,3$). That is, you get the intervals $$ [-1,\color{blue}{0}],\ [0,\color{blue}{1}],\ [1,\color{blue}{2}],\ [2,\color{blue}{3}].\ $$ Since you are doing the right endpoint method, you will be using the right endpoints of each of these intervals.

  2. Find the $y$ values of those right endpoints.

  3. You have four rectangles now, with the base and height for each.

  4. Calculate the area of each of those four rectangles (area = base $\times$ height).

  5. Add up those four areas.

You will have found an approximation for $\int_{-1}^3 f(x)\,dx$.

A picture is worth 1000 words.

  • 1
    $\begingroup$ Thanks @JohnD You really just made it so much easier to look at $\endgroup$
    – Csci319
    Dec 3, 2014 at 22:04
  • $\begingroup$ $(0,1)$, $(1,2)$, $(2,5)$, and $(3,10)$ Then you add the areas: $0+2+10+30=42$ is that correct? $\endgroup$
    – Csci319
    Dec 3, 2014 at 22:09
  • $\begingroup$ Do four "base $\times$ height" calculations: $1\cdot f(0)+1\cdot f(1)+1\cdot f(2)+1\cdot f(3)=(1)(1)+(1)(2)+(1)(5)+(1)(10)=18$. $\endgroup$
    – JohnD
    Dec 3, 2014 at 22:11

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