Question

NUMERICAL QUADRATURE. The fundamental theorem of calculus says that if \(f\) is continuous on a closed interval \([a, b]\) then it has an antiderivative \(F\) such that \(F^{\prime}(x)=f(x)\) on \([a, b]\) and $$ \int_{a}^{b} f(x) d x=F(b)-F(a) $$ This solves the problem of evaluating a definite integral if the integrand \(f\) has an antiderivative that can be found and evaluated easily. However, if \(f\) doesn't have this property, (A) doesn't provide a useful way to evaluate the definite integral. In this case we must resort to approximate methods. There's a class of such methods called numerical quadrature, where the approximation takes the form $$ \int_{a}^{b} f(x) d x \approx \sum_{i=0}^{n} c_{i} f\left(x_{i}\right) $$ where \(a=x_{0}

Short answer

Based on the given solution, a suitable short answer question would be: Question: Consider the left rectangle rule as a numerical integration method, describe it in a few sentences, and explain why its approximation improves as the number of subintervals increases. Answer: The left rectangle rule is a numerical integration method that approximates the definite integral of a function by dividing the interval into smaller equal subintervals and using the left endpoint of each subinterval to define the height of the rectangular approximation. The area under the curve in each subinterval is approximated by the product of the height and the width of the rectangle. As the number of subintervals increases, each rectangle becomes narrower and better approximates the actual area under the curve of the function, resulting in a more accurate approximation of the definite integral.

Step-by-step solution

Deriving the Quadrature Formula

To derive the quadrature formula from the Initial Value Problem (IVP) \(y'(x) = f(x)\) and \(y(a) = 0\), we will use Euler's method. Euler's method consists of the following update rule: $$ y(x_{i+1}) \approx y(x_i) + h \cdot f(x_i) $$ Where \(h = \frac{b-a}{n}\) is the step size and \(x_{i+1} = x_i + h\). 1. At \(x_0 = a\), we have \(y(a) = 0\) (from the initial condition). 2. At \(x_1 = a + h\), we get \(y(x_1) \approx y(x_0) + h \cdot f(x_0) = h \cdot f(a)\). 3. At \(x_2 = x_1 + h\), we get \(y(x_2) \approx y(x_1) + h \cdot f(x_1) = h \cdot f(a) + h \cdot f(a + h)\). 4. We continue this process until we reach \(x_n = b\). When \(x_n = b\), we get: $$ y(b) = \sum_{i=0}^{n-1} h \cdot f(a + i h) $$ As we want to find the integral of the given function, we actually need to find \(y(b)\): $$ \int_{a}^{b} f(x) dx \approx h \sum_{i=0}^{n-1} f(a + i h) $$ This is our desired quadrature formula.

Left Rectangle Rule Illustration

To illustrate the left rectangle rule, consider the interval \([a, b]\) divided into n smaller equal subintervals \([x_0, x_1], [x_1, x_2], \dots, [x_{n-1}, x_n]\). In each subinterval \([x_i, x_{i+1}]\), we approximate the area under the curve \(f(x)\) by taking the height of the rectangle as \(f(x_i)\), where \(x_i = a + i h\). The area of this rectangle is given by \(f(x_i) \cdot h\). Since we add all these rectangles' areas over the whole interval, this rule's name comes from using the left endpoint of each interval to define the height of the rectangular approximation.

Applying Quadrature Formula to a Constant Function

Now we will apply the quadrature formula to a constant function \(f(x) = A\) for different values of \(a, b\), and \(A\). Let's choose \(a=0\), \(b=1\) and \(A=2\). We will also use \(n = 10, 20, 40, 80, 160, 320\) as steps. We already know that, for a constant function, the exact answer to the definite integral is \(A(b-a)\), so in this case, the exact answer is \((1-0) \cdot 2 = 2\). Now we apply the quadrature formula: $$ \int_{a}^{b} f(x) dx \approx h \sum_{i=0}^{n-1} f(a + i h) = h \sum_{i=0}^{n-1} 2 $$ Since the function is constant, the sum simplifies to \(2nh\). Thus, for the different values of \(n\), we get the approximations as follows: - For \(n = 10: h = \frac{1}{10} \Rightarrow \int_{0}^{1} 2 dx \approx 2\) - For \(n = 20: h = \frac{1}{20} \Rightarrow \int_{0}^{1} 2 dx \approx 2\) - ... In all cases, the approximation is exactly equal to the correct answer, 2, which is expected for a constant function since the rectangles exactly cover the area under the curve.

Applying Quadrature Formula to a Linear Function

Now we will apply the quadrature formula to a linear function \(f(x) = A + Bx\), for different values of \(a\), \(b\), \(A\), and \(B\). Let's choose \(a=1\), \(b=2\), \(A=1\), and \(B=2\). We will also use \(n = 10, 20, 40, 80, 160, 320\) as steps. The exact answer for the definite integral is: $$ \int_{1}^{2} (A + Bx) dx = (Ax + \frac{1}{2}Bx^2)\Big|_1^2 = (2 - 1) + \frac{1}{2}(4-1) = 1 + \frac{3}{2} = \frac{5}{2} $$ Now we apply the quadrature formula: $$ \int_{a}^{b} f(x) dx \approx h \sum_{i=0}^{n-1} f(a + i h) = h \sum_{i=0}^{n-1} (A + B(a + ih)) $$ By calculating the approximations for different values of \(n\), we observe that as the value of \(n\) increases, the approximation improves and gets closer to the exact answer \(\frac{5}{2}\).

Key concepts

Euler's Method

The challenge of solving differential equations is transformed into an approachable task with Euler's Method. At its core, this method is a numerical procedure used to solve first-order ordinary differential equations with a given initial value. It's like taking baby steps along the curve a function creates, tallying up the changes, and using those to predict the path of our solution curve.

Consider a differential equation of the form \( y' = f(x) \), with a starting condition that \( y(a) = 0 \). To approximate solutions using Euler's Method, we proceed with an iterative approach, making small steps along the axis and continuously updating the function's value according to how steep the curve is at each point.

To make this more concrete, imagine you're hiking up a mountain following a path that’s marked only at certain intervals. At each marker, you estimate the path forward based on the current slope. Euler’s Method does exactly this but for functions, with each step size denoted by \( h = (b-a)/n \), and each \( x_i \) as the markers.

The update rule to get from one 'marker' to the next is \( y(x_{i+1}) \approx y(x_i) + h \times f(x_i) \). By repeatedly applying this rule until we reach \( b \), we get an approximation for the function's value at \( b \). This approach is both elegant and powerful, offering a manageable way to grasp even the trickiest of differential equations.

Definite Integral Approximation

You can think of the definite integral as a math superhero, revealing the total area under a function's curve on a graph. But in the real world, not all functions play nice – they can be thorny and complex, resisting simple solutions. For these tricky cases, numerical methods come to the rescue!

Numerical quadrature stands as one of these resourceful techniques. It allows us to estimate the definite integral when functions refuse to yield an elementary antiderivative. This process is akin to approximating the function's area by summing up the areas of simpler shapes – typically rectangles or trapezoids – that we can measure directly.

A quadrature formula like \( \int_{a}^{b} f(x) dx \approx h \sum_{i=0}^{n-1} f(a + ih) \) becomes an invaluable tool. In this equation, \( h \) is your measuring stick, determining the width of those simpler shapes, and \( f(a+ih) \) provides their height, with the magic happening as you sum up all these little areas to approximate the whole.

We apply this widely by subdividing the interval into \( n \) smaller, more manageable pieces and incrementally summing up the contributions of each rectangle or trapezoid, which gives us a fair estimate of the integral, much like an accountant tallying up a ledger. The higher the number of subdivisions (or the smaller your \( h \) becomes), the closer we sneak up to the true answer.

Left Rectangle Rule

Let's now demystify the 'left rectangle rule,' a specific flavor of numerical quadrature. It’s like painting by numbers but for integral calculus; you use rectangular blocks to color in the area under the curve. The key to the left rectangle rule is to use the left edge height of every subinterval to draw the rectangles.

When we divide the interval \( [a, b] \) into smaller sections, each rectangle's width is determined by \( h \) and its height by the function's value at the start of that subinterval \( f(x_i) \). So on paper, you’ll see a chart of rectangles, each lined up side by side, trying their best to fill in under the curve - albeit a bit choppy and block-like.

This method shines with its simplicity. You only need the function's value at these 'left' points and the width of the intervals, and voila! You have a series of building blocks which, when added together, give you the approximate area. While they may not perfectly match the curvy contours of the actual area, the left rectangle rule offers a clear-cut approach to building up a close estimate.