Estimate the area under the curve f(x)=x^2 between x=0 and x=1 using 4 rectangles of equal width if
a) The height is taken from the right endpoint
b) The height is taken from the midpoint.
I have no idea what exactly this question is asking me to do is this suppose to be a bar graph?
You want to find the area under the curve. The easy thing for doing area is a rectangle: width*height.
So, divide the interval into 4 sections of equal width, and use y as the height. But which y? on the left side, right side, in the middle, or what? Doesn't matter; pick one and try it.
If you have no idea what is wanted, you haven't listened in class, or read the material, or both. Or (gasp!) gone online to study up on the topic.
Below are tables showing the iteration number, x,y values, and partial sum.
RIGHT-HAND RULE:
1: (0.2500,0.0625) 0.01562
2: (0.5000,0.2500) 0.07812
3: (0.7500,0.5625) 0.21875
4: (1.0000,1.0000) 0.46875
MIDPOINT RULE:
1: (0.1250,0.0156) 0.00391
2: (0.3750,0.1406) 0.03906
3: (0.6250,0.3906) 0.13672
4: (0.8750,0.7656) 0.32812
Actual value: 0.33333
No, this question is not asking you to create a bar graph. It is asking you to estimate the area under the curve of the function f(x) = x^2 between x=0 and x=1 using a method called "rectangle approximation." This method involves approximating the area under a curve by dividing it into multiple rectangles and then summing up the areas of these rectangles.
To estimate the area under the curve using rectangle approximation, you need to follow these steps:
1. Find the width of each rectangle: Since you are given that there are 4 rectangles of equal width, divide the total width (1 - 0 = 1) by the number of rectangles (4). In this case, the width of each rectangle would be (1 - 0) / 4 = 0.25.
2. Determine the height of each rectangle: Depending on whether the height is taken from the right endpoint or the midpoint, the procedure will differ.
a) Height taken from the right endpoint: For this approach, you will evaluate the function at the right endpoint of each rectangle, which is simply the x-coordinate of the right side of each rectangle. In this case, the rectangles will have endpoints at x=0.25, x=0.5, x=0.75, and x=1. The heights of the rectangles will be f(0.25), f(0.5), f(0.75), and f(1), which can be calculated as 0.0625, 0.25, 0.5625, and 1, respectively.
b) Height taken from the midpoint: For this approach, you will evaluate the function at the midpoint of each rectangle, which is the average of the x-coordinates of the left and right sides of each rectangle. In this case, the midpoints of the rectangles will be x=0.125, x=0.375, x=0.625, and x=0.875. The heights of the rectangles will be f(0.125), f(0.375), f(0.625), and f(0.875), which can be calculated as 0.015625, 0.140625, 0.390625, and 0.765625, respectively.
3. Calculate the area of each rectangle: To find the area of each rectangle, multiply the width of the rectangle by its height.
4. Sum up the areas of all the rectangles: Finally, add up the areas of all the rectangles to estimate the area under the curve. So, for both cases (a and b), simply sum up the areas calculated in Step 3.
By following these steps, you can estimate the area under the curve f(x) = x^2 using rectangle approximation with either right endpoints or midpoints to determine the heights.