91Ó°ÊÓ

A Riemann sum is a method for approximating the total value of a function over an interval. It does this by breaking the interval into smaller sub-intervals or partitions. For each sub-interval, you take a sample point, calculate the function's value there, and multiply it by the width of the sub-interval. This value approximates the area under the curve for that segment.

By adding the areas of all these sub-intervals together, you get the Riemann sum, which approaches the integral as the width of sub-intervals approaches zero.

In the exercise, the expression \( \sqrt{4-c_k^2} \, \Delta x_k \) represents the Riemann sum for the function \( f(x) = \sqrt{4-x^2} \) over the interval \([0,1]\). As \(|P| \rightarrow 0\), this sum becomes a more accurate approximation of the definite integral of the function over the interval.

Intervals are used to describe a set of real numbers lying between two endpoints. Interval notation is a handy tool for expressing such sets concisely and clearly.

The notation has two forms:

• Closed interval \([a, b]\), where both \(a\) and \(b\) are included in the set.
• Open interval \((a, b)\), where \(a\) and \(b\) are not included.

The given Riemann sum in the exercise is defined over the closed interval \([0, 1]\). This tells us that the integration includes both the starting number, 0, and the ending number, 1.

Function representation is the process of expressing a mathematical function with an equation, graph, or other means. In calculus, functions describe relationships between two variables, typically \(x\) and \(y\).

The function presented in the exercise is \(f(x) = \sqrt{4-x^2}\). This is a semicircle with a radius of 2 and is part of the equation of a circle \(x^2 + y^2 = 4\). This particular function only describes the upper half of the circle. By examining this function over the interval \([0,1]\), we look at part of this semicircle and calculate the area beneath it for that portion.

Integral calculus is a branch of calculus focused on finding the total accumulation of a quantity. For functions, this often involves calculating the area under a curve.

The definite integral, denoted by \(\int_{a}^{b} f(x) \, dx\), represents the accumulation of the function \(f(x)\) from \(x = a\) to \(x = b\). Definite integrals combine limits and summation to calculate precise quantities.

In the exercise, the Riemann sum is transformed into a definite integral, \(\int_{0}^{1} \sqrt{4-x^2} \, dx\). This represents the exact area under the curve of \(f(x) = \sqrt{4-x^2}\) from \(x = 0\) to \(x = 1\). It is a crucial technique in calculus for solving real-world problems, where precise calculations are necessary.