Question

Evaluate the limit and sketch the graph of the region whose area is represented by the limit. \(\lim _{\| \Delta \rightarrow 0} \sum_{i=1}^{n}\left(4-x_{i}^{2}\right) \Delta x,\) where \(x_{i}=-2+(4 i / n)\) and \(\Delta x=4 / n\)

Short answer

The value of the limit is \(\frac{32}{3}\), which is the area of the region under the curve of the function \(f(x) = 4 - x^2\) from -2 to 2.

Step-by-step solution

Recognizing the Function and Framework

Identify the function inside the sum, \(4 - x_i^2\), and recognize that the limit is the definition of a Riemann sum. A Riemann sum is a certain kind of approximation of an integral by a finite sum.

Finding Expressions for Limits

Express the lower and upper limits of the integral in terms of \(x_i\) and \(\Delta x\). Here, the lower and upper limits of x can be obtained from the equation given for \(x_i\), i.e., \(x_i = -2 +\frac{4i}{n}\). When \(i = 1\), \(x_1 = -2 + \frac{4(1)}{n} = -2 + \frac{4}{n}\). And when \(i = n\), \(x_n = -2 + \frac{4n}{n} = -2 + 4 = 2\). Thus, the limits for \(x\) are from -2 to 2.

Evaluating The Integral

Now, evaluate the definite integral from -2 to 2 of \(f(x) = 4 - x^2\) using the fundamental theorem of calculus. The anti-derivative of \(4 - x^2\) is \(4x - \frac{x^3}{3}\). Evaluating it from -2 to 2, we get \([8 - \frac{8}{3}] - [-8 - \frac{8}{3}] = \frac{32}{3}\).

Sketching the Graph

The function \(f(x) = 4 - x^2\) is a simple quadratic function that opens downwards (since the coefficient of \(x^2\) is negative) with the vertex at the point (0, 4). It intersects the x-axis at x = -2 and x = 2 (the roots of the equation). The area under the curve between x = -2 and x = 2 obtained from the integral is the limit of the Riemann sum.

Key concepts

Definite Integral

The concept of a definite integral is fundamental to understanding calculus. It represents the exact area under the curve of a function on a specific interval. In our exercise, we dealt with the area under the quadratic function, represented by the equation \(4 - x^2\), from \(x = -2\) to \(x = 2\). To find this, we determined the anti-derivative, also known as the indefinite integral, which reverses differentiation. We then applied the boundaries of our interval to this anti-derivative to get the exact area, which is the essence of the definite integral.

Evaluated mathematically, the integral becomes \[\int_{-2}^{2} (4-x^2) dx\], which yields the exact number \(\frac{32}{3}\), representing that specific area. Definite integrals are incredibly useful for finding not just geometric areas, but also for solving physical problems involving quantities like work, probability distributions, and cumulative growth.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two principal concepts in calculus. This theorem has two parts: the first part gives us an anti-derivative of a function, and the second part relates the definite integral of a function to its antiderivative.

Regarding our exercise, we applied the second part by calculating the antiderivative of our function to evaluate the definite integral. The theorem states that if a function \(f\) is continuous on a closed interval \[a, b\], then the function has an antiderivative on that interval, and the integral of \(f\) from \(a\) to \(b\) is equal to the antiderivative evaluated at \(b\) minus the antiderivative evaluated at \(a\). This result allows us to compute the area under curves easily and it is beautifully exemplified in the solution to our exercise where \[\int_{-2}^{2} (4-x^2) dx\] resulted in \(\frac{32}{3}\).

Limit of a Function

The limit of a function is a foundational concept in calculus, addressing the behavior of functions as they approach a particular point. We often talk about the limit of a sequence or function as it approaches some value or infinity.

In the exercise, the limit sign in front of the Riemann sum \(\lim_{\Delta x \rightarrow 0} \sum_{i=1}^{n}(4-x_i^2) \Delta x\) indicates that we look at what happens as the width of the subintervals \(\Delta x\) gets smaller and smaller. Effectively, we're determining the area under the curve by an ever-increased number of rectangles whose total area approaches the integral of the function – this is the precise definition of the definite integral.

Area Under a Curve

Understanding the area under a curve is crucial in many fields of mathematics and science. It allows for the quantitative analysis of spaces bounded by the graph of a function and the axes on a coordinate plane.

In our exercise's context, by computing the area under the curve of \(f(x) = 4 - x^2\) between \(x = -2\) and \(x = 2\), we essentially found the integral of the function over that range. The Riemann sum approximation using rectangles acts as stepping stones that lead us to the exact area as their width diminishes to zero, which is the area represented by the definite integral. Our final calculation gives us a numerical value for this area, \(\frac{32}{3}\), a precise outcome that ties together several concepts of calculus.