# Estimate the area under the graph

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

• 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. Dec 3, 2014 at 21:55
• Wolfram Alpha is really powerful nowadays. :) Dec 3, 2014 at 22:53
• @anorton I know but I want to learn how to do this problem and sometimes Wolfram Alpha does not show you Dec 3, 2014 at 22:56
• @BS319 True; I thought the picture was kind-of nice, though. :) Dec 3, 2014 at 22:57

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.

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