Approximate the area under the curve y=x3 from x=0 to x=2 using a Right Endpoint approximation with 4 subdivisions.

Answer 1

Set up a table with x = {0, 0.5, 1.0, 1.5, 2.0} for which y = {0, 0.125, 1.0, 3.375, 8.0}. After the first y value of 0, the rest are right endpoints of 4 subdivisions. Add them up and divide by the width of the interval [0,2] to get 12.5/2 = 6.25.

Answer 2

Great question! This relates really closely with a huge concept in calculus called integration.

Let's work the problem statement:

1. 4 subdivisions means breaking the interval from x = 0 to x = 2 into 4 pieces. Since 2/4 = 0.5, each interval will have a width of 0.5.
2. Right Endpoint means we're looking for the right side of each of those 0.5 intervals. That means we're interested in something about 0.5, 1, 1.5, and 2.
3. approximation is describing using rectangles to estimate the area under the curve. It's not perfect, but it will give us a number somewhat close. So we simply need the height of the 4 rectangles multiplied by the width of 0.5. This particularly answer will be an overestimate, because the curve is increasing on the interval in question.)
4. the curve y=x3 means we're interested in the values of x³ at each of these points. These values are 0.125, 1, 3.375, and 8.

From there, we simply add up the areas of the rectangles, 0.5 times each of those numbers:

0.5*0.125 + 0.5*1 + 0.5*3.375 + 0.5*8 = 6.25 (or 25/4).

Note: You're approximating:

https://www.wolframalpha.com/input/?i=int%28x%5E3%2C0%2C2%29

using:

sum((0.5*x)^3,x,1,4)*0.5