Consider the area between the curve y=x^2 and the x-axis over the interval 0<=x<=1. Assume n is a whole number. use Fermat's method of integration to show that this area will equal 1/(n+1).
thank you
To calculate the area between the curve y=x^2 and the x-axis over the interval 0<=x<=1 using Fermat's method of integration, we need to follow the following steps:
Step 1: Divide the interval [0, 1] into n equal subintervals.
In this case, we have the interval [0, 1], and we need to divide it into n equal subintervals. Each subinterval will have a width of Δx = (1-0)/n = 1/n.
Step 2: Select representative points for each subinterval.
We need to select a representative point for each subinterval. In Fermat's method of integration, we choose the x-coordinate of the left endpoint of each subinterval as the representative point.
Step 3: Calculate the sum of the areas of the rectangles.
For each subinterval, we construct a rectangle with a width of Δx and a height of f(xi), where xi is the representative point for that subinterval. In this case, the height will be f(xi) = (xi)^2 = (xi)^2.
Step 4: Find the limit of the sum as n approaches infinity.
To find the area between the curve and the x-axis, we need to find the limit of the sum of the areas of the rectangles as the number of subintervals approaches infinity. Mathematically, this is represented as the limit of the Riemann sum:
lim(n→∞) Σ[i=1 to n] [(xi)^2 * Δx]
Step 5: Evaluate the limit.
To evaluate the limit, we need to find the definite integral of the function f(x) = x^2 over the interval [0, 1]. The definite integral represents the area between the curve and the x-axis. So, we have:
∫[0,1] (x^2) dx = [x^3/3] from 0 to 1 = 1/3 - 0/3 = 1/3
Step 6: Simplify the result.
Now, we need to simplify the result in terms of n. We know that the limit of the Riemann sum, as n approaches infinity, is the definite integral mentioned earlier. So, we have:
lim(n→∞) Σ[i=1 to n] [(xi)^2 * Δx] = ∫[0,1] (x^2) dx = 1/3
Since the area between the curve and the x-axis is equal to 1/3, it means that the sum of the areas of the rectangles is also equal to 1/3.
Step 7: Express the result in terms of n.
Now, we need to express the sum of the areas of the rectangles in terms of n. Since the sum is equal to 1/3, we have:
Σ[i=1 to n] [(xi)^2 * Δx] = 1/3
Step 8: Rearrange the equation and solve for n.
We can rearrange the equation to solve for n:
1/3 = Σ[i=1 to n] [(xi)^2 * Δx]
1/(n+1) = (1/3) * (1/n)
Therefore, using Fermat's method of integration, we have shown that the area between the curve y=x^2 and the x-axis over the interval 0<=x<=1 is equal to 1/(n+1).