Final answer:
The integral of x from 0 to 1 is calculated using the definition of the definite integral, which involves taking the limit of a Riemann sum. The final value of the integral ∫¹₀ x dx is ½.
Step-by-step explanation:
Computing the Definite Integral Using the Definition
To compute the integral ∫¹₀ x dx, we start by considering the definition of a definite integral. This involves taking the limit of a Riemann sum as the partition of the interval [0,1] gets finer. The integral represents the area under the curve of the function f(x) = x from x = 0 to x = 1.
First, we partition the interval from 0 to 1 into n equal subintervals, each of length Δx = 1/n. Then, we will choose a sample point within each subinterval, typically the right end point, which we can denote as xᵢ where i ranges from 1 to n. The Riemann sum for this partition is:
Sₙ = ∑_{i=1}^{n} xᵢ * Δx = ∑_{i=1}^{n} ɒ Δx
As n becomes infinitely large, Δx tends to zero and the Riemann sum approaches the actual area under the curve:
lim_{n→∞} Sₙ = ∫¹₀ x dx
Substituting in the value for xᵢ and Δx, we get:
Sₙ = ∑_{i=1}^{n} (i/n) * (1/n) = (1/n²) * ∑_{i=1}^{n} i
The sum of the first n integers is ½n(n+1), so the Riemann sum simplifies to:
Sₙ = (1/n²) * ½n(n+1) = (n+1)/2n
Finally, taking the limit as n approaches infinity we find the value of the integral:
lim_{n→∞} Sₙ = lim_{n→∞} (n+1)/2n = ½
Therefore, the value of the integral ∫¹₀ x dx is ½.