Question

In Exercises 1-6, (a) use the Trapezoidal Rule with n = 4 to approximate the value of the integral. (b) Use the concavity of the function to predict whether the approximation is an overestimate or an underestimate. Finally, (c) find the integral's exact value to check your answer. $$\int_{0}^{2} x d x$$

Short answer

The trapezoidal rule approximation of the integral is 4.5, which, since the function is a straight line and has neither concavity nor convexity, provides neither an overestimate nor an underestimate. The exact value of the integral is 2.

Step-by-step solution

Apply the Trapezoidal Rule

The trapezoidal rule for approximating an integral over interval [a,b] is given as \[I = \frac{h}{2} [y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n]\] where \(h = \frac{b-a}{n}\), \(y_i = f(a + ih)\), and n = number of trapezoids. For our case, \(f(x) = x\), a = 0, b = 2, and n = 4, so \(h = \frac{2-0}{4} = 0.5\). Hence, \(y_0 = f(0) = 0\), \(y_1 = f(0.5) = 0.5\), \(y_2 = f(1) = 1\), \(y_3 = f(1.5) = 1.5\), \(y_4 = f(2) = 2\). Using these in the trapezoidal rule formula: \[I = \frac{0.5}{2} [0 + 2(0.5 + 1 + 1.5) + 2] = 1 + 2.5 + 1 = 4.5\]

Discuss the concavity

This function \(f(x) = x\) is a straight line, thus it has neither concavity nor convexity. Consequently, the trapezoidal rule will not overestimate nor underestimate.

Find the exact integral

The exact value of the integral can be found by evaluating the antiderivative of \(f(x) = x\) at the limits of integration. The antiderivative of \(x\) is \(\frac{1}{2}x^2\). So, the exact integral is: \[\int_0^2 x dx = \frac{1}{2} * 2^2 - \frac{1}{2} * 0^2 = 2 - 0 = 2\]

Key concepts

Numerical Integration

Numerical integration is a cornerstone of computational mathematics, helping us approximate the values of definite integrals where an analytic solution is difficult or impossible to obtain. It uses algorithms to calculate an approximate answer to the area under a curve or, more formally, the integral of a function over a certain interval. The Trapezoidal Rule is one of these algorithms, balancing simplicity and precision by dividing the area into trapezoids rather than more complex shapes. Let's break down this process further:
When applying the Trapezoidal Rule, we mimick the curve of a function using straight line segments, creating trapezoids under these segments. The sum of the areas of these trapezoids gives us an approximate value of the integral. The accuracy of this method hinges on the number of trapezoids (

• ) — generally, more trapezoids mean a more accurate approximation but at the cost of more complex calculation.
  In the example of , with four trapezoids, we obtain the approximate integral value of

.For functions that can't be easily integrated analytically, such as those with no elementary antiderivatives, numerical integration becomes an invaluable tool.

Definite Integrals

Definite integrals are all about finding the exact area under a curve between two boundaries. Mathematically, this is denoted as , where f(x) is the function, and a and b are the lower and upper limits of integration respectively. Definite integrals are key to understanding a multitude of phenomena such as distances, areas, and total quantities.In the exercise , the definite integral represents the exact area under the line f(x) = x from x = 0 to x = 2. Since the function f(x) = x is a straight line with a constant slope, its integral can be thought of as the area of a single trapezoid, making the computation straightforward. The exact area—or the definite integral—comes out to be which aligns well with our intuition of calculating the area of a triangle with base 2 and height 2.

Concavity and Curve Estimation

Concavity refers to the curvature of a function: whether it curves upwards (concave up) or downwards (concave down). This concept is crucial when using numerical methods like Trapezoidal Rule because the rule tends to overestimate the area under curves that are concave down and underestimate those that are concave up.Curve estimation, then, is the process of determining how our approximation might differ from the exact value based on the concavity of the function over the interval. In the example from the exercise, the function f(x) = x is a straight line with no concavity. This unique case makes the Trapezoidal Rule’s estimate exactly accurate as the 'trapezoids' formed are actually rectangles without any slant, perfectly matching the actual shape under the graph.

Antiderivatives

Antiderivatives, or indefinite integrals, represent the reverse process of differentiation. If we have a function f(x), its antiderivative F(x) is a function that, when differentiated, yields f(x). For example, if f(x) = x, then an antiderivative of f is F(x) = \(\frac{1}{2}x^2 + C\), where C is a constant. Knowing the antiderivative is essential in computing the exact value of definite integrals.By evaluating F(x) at the upper and lower limits of integration, we can find the exact area under the curve between these limits, known as the Fundamental Theorem of Calculus. In our exercise, the antiderivative of f(x) = x is used to evaluate the exact value of the integral, resulting in an area of 2. This highlights the crucial role antiderivatives play in finding exact solutions to definite integrals—a concept deeply interwoven with the idea of finding the area under a curve across a given interval.