Question

\(f(x)\) is given by \(x: 0\) 1 \(0.5\) \(0.5\) 1 \(f(x)\) 1 \(0.8\) then using Trapezoidal rule, the value of \(\int_{0}^{1} f(x) d x\) is ...

Short answer

0.65

Step-by-step solution

Understand the Trapezoidal Rule

The Trapezoidal Rule is a numerical method for estimating the definite integral of a function. The basic idea is to approximate the region under the graph of the function as a trapezoid and calculate its area. The formula for the Trapezoidal Rule with two points, \(a\) and \(b\), and the function values at these points, \(f(a)\) and \(f(b)\), is \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2} \left[ f(a) + f(b) \right] \].

Identify the Points and Function Values

From the given table, we have two points: \((x_0, f(x_0)) = (0, 0.5)\) and \((x_1, f(x_1)) = (1, 0.8)\). These correspond to \(a = 0\) and \(b = 1\), with the function values \(f(0) = 0.5\) and \(f(1) = 0.8\).

Apply the Trapezoidal Rule Formula

Substitute the identified values into the formula: \[ \int_{0}^{1} f(x) \, dx \approx \frac{1 - 0}{2} \left[ f(0) + f(1) \right] = \frac{1}{2} \left[ 0.5 + 0.8 \right] \].

Compute the Approximate Integral

Calculate the expression from Step 3: \[ \frac{1}{2} (0.5 + 0.8) = \frac{1}{2} \times 1.3 = 0.65 \].

Interpret the Result

The approximate value of the integral \( \int_{0}^{1} f(x) \, dx \) using the Trapezoidal Rule is \(0.65\).

Key concepts

Numerical Integration

Numerical Integration is a technique used to approximate the value of a definite integral, especially when an exact form of the integral cannot be easily determined. It provides useful methods for finding integrals in practical scenarios where functions may be complex or only available as discrete data points. Unlike symbolic integration, which finds the exact integral analytically, numerical integration involves approximation methods.

There are various methods under the umbrella of numerical integration, such as:

• Trapezoidal Rule
• Simpson's Rule
• Midpoint Rule

Each of these methods has its strengths and is chosen based on the requirements of the problem and the function characteristics. In particular, the Trapezoidal Rule is straightforward and intuitive, making it popular for simple to moderately complex functions.

Definite Integral

A Definite Integral of a function gives the area under the curve of that function, between specified limits on the x-axis, from a lower bound to an upper bound. Mathematically, it is represented as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the limits of integration.

Definite integrals are used in various fields such as physics, engineering, and economics to determine quantities like:

• Area under a curve
• Accumulated change in a quantity
• Total profit or loss over a given interval

Visually, it can be seen as the area between the curve of \(f(x)\), the x-axis, and the vertical lines \(x = a\) and \(x = b\). By using numerical methods like the Trapezoidal Rule, we can estimate this area even if we can't compute it exactly through analytical means.

Trapezoid Area Approximation

Trapezoid Area Approximation is the basis of the Trapezoidal Rule, where the area under a curve is estimated by dividing it into a series of trapezoids rather than rectangles, which are used in simpler methods like the Riemann sum. Each trapezoid provides a better approximation of the area beneath a curve between two points than rectangles as it accounts for the slope of the function.

For a single interval \([a, b]\), the area of the trapezoid is given by: \[ \frac{b-a}{2} \left[ f(a) + f(b) \right] \]
This formula comes from averaging the function's values at the interval's endpoints and multiplying by the interval's width. The main advantage is that the trapezoid settles closely along the path of the curve, providing a more accurate approximation compared to methods using rectangles. When multiple intervals are used over a curve, the total area is the sum of all individual trapezoid areas, yielding good approximations for a variety of functions.