The Riemann sum s for f(x)=4x^2, 0<=x<=1, taking the sample points to be the right endpoints is given by s=4n^2+6n+2/3n^2. True or False?
there are many handy Riemann um calculators online, such as
http://www.emathhelp.net/calculators/calculus-2/riemann-sum-calculator/
To determine if the given expression for the Riemann sum s is true or false, let's analyze it step by step.
The Riemann sum for a function f(x) over an interval [a, b] using n subintervals can be calculated by evaluating the following expression:
s = (b - a) / n * Σ f(xi),
where Σ denotes the summation notation, xi represents the sample points within each subinterval, and (b - a) / n represents the width of each subinterval.
In this case, the function f(x) is given as f(x) = 4x^2, and the interval is [0, 1]. We are instructed to take the sample points as the right endpoints.
The width of each subinterval is (1 - 0) / n = 1 / n, as the interval spans from 0 to 1.
To find the right endpoints, we need to partition the interval into n subintervals. The endpoints of the subintervals will be: x1 = 1/n, x2 = 2/n, x3 = 3/n, and so on, up to xn = n/n = 1.
Now, let's calculate the Riemann sum using the given expression:
s = (1 / n) * Σ f(xi).
Since the sample points are the right endpoints, the expression becomes:
s = (1 / n) * Σ f(i/n),
where i ranges from 1 to n.
Substituting f(x) = 4x^2 into the equation:
s = (1 / n) * Σ 4(i/n)^2.
To simplify, let's manipulate the expression further:
s = (1 / n) * 4 * Σ (i^2) / n^2,
simplifying to:
s = (4 / n^3) * Σ (i^2).
Now, we can observe that the summation Σ (i^2) is equivalent to the sum of the squares of the first n natural numbers, which can be calculated using the formula:
Σ (i^2) = n * (n + 1) * (2n + 1) / 6.
Plugging this result back into the expression, we have:
s = (4 / n^3) * [n * (n + 1) * (2n + 1) / 6].
Simplifying further:
s = (2 / 3) * (n + 1) * (2n + 1) / n^2.
Comparing this expression to the given expression for s:
s = 4n^2 + 6n + (2 / 3n^2).
We can see that the given expression is NOT equivalent to the calculated expression for s.
Therefore, the statement "s = 4n^2 + 6n + (2 / 3n^2)" is FALSE.