Estimate the area under $f(x) = \frac{1}{2}x^2 - 1$ from $x = 0$ to $4$ using two right hand rectangles.

Question

Help with beginner college calc plz:
I can't upload pictures on my computer (it's a very old desktop) but I'll try and explain what I did. So obviously I started with graphing f(x);

(2, 1)
(-2, 1)
(0, -1)
(3, 4)
(-3, 4)

Then I drew my rectangles. The first one's width was 0 to 1.5, and the 2nd's width was 1.5 to 3 (this is where my problem came in - I will explain). My height for the 1st rectangle was 2 and the 2nd was 4. To reiterate these are right hand rectangles.

Then I did l . w so the 1st rectangle's area would be 3 and the 2nd's 12 adding up to 15.

My question though is how would you draw your two rectangles? The estimate should be from 0 to 4, but my 2nd rectangle's upper right corner only touched 3,4.

Edit: Or did I graph my parabola completely wrong?

Answer 1

Your graph should look like this:

Also note that $f(4)=7$ and $f(3)=\frac{1}{2}\cdot 9-1$.