91Ó°ÊÓ

When dividing an interval into subintervals, the idea is to break the whole range into smaller, equal parts. Consider the interval \([a, b]\). We divide this interval into \(n\) equal parts, each with a width or length called \(\Delta x\). This width is calculated as \(\Delta x = \frac{b-a}{n}\).
In our specific exercise, the interval is \([0, 1]\), and the number of divisions or subintervals is \(n\). So, the width becomes \(\Delta x = \frac{1}{n}\). Dividing the interval like this allows us to make approximate calculations for functions, which form the base for defining a Riemann sum.

After deciding on the subintervals, the next step involves choosing points within these subintervals for calculation purposes. A common choice is the right-hand endpoint of each subinterval.
This is identified as the point which is at the right edge of each interval division. If our interval spans from \(a\) to \(b\), and is divided into \(n\) subintervals, the right-hand endpoint for each subinterval is \(c_k = a + k \Delta x\).

• For \([0, 1]\) with equal parts, this becomes \(c_k = \frac{k}{n}\)
• Here, \(k\) runs from 1 to \(n\), covering every portion within the main interval from the right-hand side.

This selection is crucial as it determines where in the function we are evaluating to approximate the area under the curve.

Calculating the limit is an essential part of determining the Riemann sum accurately as the number of subintervals approaches infinity.
To find the exact area under a curve through Riemann sums, we use the concept of taking limits. This implies letting \(n\rightarrow \infty\).
By doing so, each subinterval's width \(\Delta x\) will shrink down to nearly zero, paving the way to achieving a highly precise estimation of the total area under the curve. The summation formula involves terms with \(k\) and \(k^2\) whose limits must be evaluated. As \(n\) increases, these equations simplify: \[ \lim_{n \rightarrow \infty} \left( \frac{3(n+1)}{2n} + \frac{4}{6} \right) = 2. \] This results in the determination of the exact area under the curve.

Finding the area under a curve involves integrating the function across the given interval. Instead of using calculus through indefinite integrals, we sometimes apply the Riemann sum method.
After calculating the limit of the Riemann sums, we achieve the area sought under the curve over specified bounds. In our example, to find the area under the function \(f(x) = 3x + 2x^2\) over \([0, 1]\), we perform the steps of the Riemann sum leading up to our limit calculation.
This concluded with obtaining \( \lim_{n \rightarrow \infty} S_n = 2 \).

• The area calculated via this limit gives a practical application of integration.
• It highlights how granularity in approximations yields accurate results.

Thus, the area under the curve \(f(x)\) for the given interval computes to 2, demonstrating the power of Riemann sums.