Evaluate the integral S (range from 1 to 3) (x^2 +1)dx by computing the limit of Reimann Sum.
To evaluate the integral S (range from 1 to 3) (x^2 + 1)dx using the limit of Riemann sums, we can follow these steps:
Step 1: Divide the interval [1, 3] into smaller subintervals.
Choose a value for the number of subintervals, n. The larger the value of n, the more accurate your answer will be.
Step 2: Calculate the width of each subinterval.
The width of each subinterval is given by (b - a) / n, where a = 1 (the lower limit of the interval) and b = 3 (the upper limit of the interval).
In this case, the width of each subinterval is (∆x) = (3 - 1) / n = 2/n.
Step 3: Determine the sample points in each subinterval.
There are different methods to choose the sample points within each subinterval. The most commonly used method is the right endpoint method, where we evaluate the function at the right endpoint of each subinterval.
In this case, the right endpoint of each subinterval is given by xi = a + i * ∆x, where i = 1, 2, 3, ..., n.
So, the sample points for this integral are: x1 = 1 + 1 * (2/n), x2 = 1 + 2 * (2/n), ..., xn = 1 + n * (2/n).
Step 4: Compute the Riemann sum.
The Riemann sum is given by the sum of the products of the function evaluated at each sample point and the width of the subinterval.
Rn = Σ [f(xi) * (∆x)], where the summation is from i = 1 to n.
In this case, the Riemann sum is:
Rn = Σ [(xi^2 + 1) * (∆x)] = [(x1^2 + 1) * (∆x)] + [(x2^2 + 1) * (∆x)] + ... + [(xn^2 + 1) * (∆x)]
Step 5: Take the limit as n approaches infinity.
To get the exact value of the integral, we need to calculate the limit of the Riemann sum as the number of subintervals approaches infinity.
lim(n→∞) Rn = lim(n→∞) Σ [(xi^2 + 1) * (∆x)]
By taking this limit, we are essentially summing an infinite number of infinitely small terms, resulting in the exact value of the integral.
Thus, to evaluate the integral using the limit of Riemann sums, you need to calculate the limit of the Riemann sum as the number of subintervals approaches infinity.