Use three rectangles of equal width and the left endpoint approximation method to estimate the area enclosed by the x-axis, the lines x = 1 and x = 4, and the curve y = x^(1/2).
Could you please help me understand how to do this?
Thanks in advance!
ok, the three rectangles run at x=1, 2, 3 for the left endponits, at y=x^.5
the width is 1, so the area is 1*x^.5
areas:
first 1-2, area=1*1^.5 left endpoint
second 2-3, area=1*2^.5
third 3-4, area=1*3^.5
so the sum of areas is
1+sqrt2 + sqrt3
Sure! To estimate the area enclosed by the x-axis, the lines x = 1 and x = 4, and the curve y = x^(1/2) using the left endpoint approximation method, we can divide the region into three rectangles of equal width.
Here are the steps to follow:
1. Determine the width of each rectangle: Since we want three rectangles of equal width, we need to find the total width of the region first. In this case, the region is between x = 1 and x = 4. So, the total width is given by 4 - 1 = 3. Divide this total width by the number of rectangles, which is 3. Therefore, each rectangle will have a width of 3/3 = 1.
2. Identify the heights of each rectangle: To determine the height of each rectangle, we need to evaluate the function y = x^(1/2) at the left endpoint of each interval. The intervals are [1, 2], [2, 3], and [3, 4]. The left endpoints of these intervals are 1, 2, and 3, respectively. Calculate the function values at these points:
- For x = 1, y = 1^(1/2) = 1.
- For x = 2, y = 2^(1/2) = sqrt(2).
- For x = 3, y = 3^(1/2) = sqrt(3).
3. Compute the area of each rectangle: To find the area of each rectangle, multiply the width of each rectangle by its corresponding height.
- For the first rectangle: Area = 1 (width) * 1 (height) = 1.
- For the second rectangle: Area = 1 (width) * sqrt(2) (height) = sqrt(2).
- For the third rectangle: Area = 1 (width) * sqrt(3) (height) = sqrt(3).
4. Find the total estimated area: Sum up the areas of the three rectangles to get the total estimated area. In this case, the total area is 1 + sqrt(2) + sqrt(3).
Therefore, using the left endpoint approximation method, the estimated area enclosed by the x-axis, x = 1, x = 4, and y = x^(1/2) is 1 + sqrt(2) + sqrt(3).