Question

Exact Trapezoid Rule Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.

Short answer

Question: Prove that the trapezoid rule is exact when we approximate the definite integral of a linear function. Answer: The trapezoid rule is exact for linear functions because the difference between the exact integral of the linear function and the trapezoid rule approximation is zero, as shown in the step by step solution above.

Step-by-step solution

Define linear function and trapezoid rule

Let us consider a linear function f(x) = ax + b, where a and b are constants. We aim to integrate this function over the interval [x_0, x_1], using the trapezoid rule. The trapezoid rule is a numerical integration method that approximates the definite integral of a function by calculating the area of a trapezoid. It is defined as T(f) = \frac{(f(x_0) + f(x_1))}{2} (x_1 - x_0), where f(x_0) and f(x_1) are the function values at the endpoints of the interval [x_0, x_1].

Apply the trapezoid rule to the linear function

Substituting f(x) into the expression for T(f), we get T(f) = \frac{1}{2}(ax_0 + b + ax_1 + b)(x_1 - x_0). Now we can simplify this expression: T(f) = \frac{1}{2}(a(x_0 + x_1) + 2b)(x_1 - x_0).

Compute the definite integral of the linear function

Let's calculate the definite integral of f(x) = ax + b over the interval [x_0, x_1]: \int_{x_0}^{x_1} (ax + b) dx. Using the rules of integration, this becomes \left[\frac{1}{2}ax^2 + bx\right]_{x_0}^{x_1}.

Calculate the difference between the trapezoid rule and the definite integral

We now substitute the limits of integration into the expression derived in Step 3: A_\text{exact}(f) = \frac{1}{2}ax_1^2 + bx_1 - \frac{1}{2}ax_0^2 - bx_0. It's time to compare the trapezoid rule approximation T(f) with the exact integral A_\text{exact}(f): A_\text{exact}(f) - T(f) = \frac{1}{2}(a(x_1^2 - x_0^2) + 2b(x_1 - x_0)) - \frac{1}{2}(a(x_0 + x_1) + 2b)(x_1 - x_0). Simplify this difference: A_\text{exact}(f) - T(f) = \frac{1}{2}(ax_1^2 - ax_0^2 + 2b(x_1 - x_0) - a(x_0 + x_1)(x_1 - x_0)). By factoring and using the property (a - b)(a + b) = a^2 - b^2, we find A_\text{exact}(f) - T(f) = 0.

Conclusion

Since the difference between the exact integral of the linear function and the trapezoid rule approximation is zero, we can conclude that the trapezoid rule is exact for linear functions.

Key concepts

Linear Functions

Linear functions are one of the simplest types of mathematical functions you will encounter. A linear function can be written in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. Here, \( a \) represents the slope of the line, and \( b \) is the y-intercept, the point at which the line crosses the y-axis.
Linear functions graph as straight lines on the Cartesian plane.
They are always predictable, with a constant rate of change, making them easier to work with compared to other, more complex functions.

In the context of the trapezoid rule, these properties of linear functions allow for the trapezoid rule to yield exact results. An exact result means there is no approximation error. For linear functions, integration via the trapezoid rule or analytically computing the definite integral using calculus will yield the same result.

Numerical Integration

Numerical integration is a technique used when calculating the integral of a function is difficult or impossible to do exactly. Rather than solving the integral analytically, we can approximate its value using methods like the trapezoid rule.
This approach is helpful especially when dealing with complex functions or when numerical solutions are preferable.

The trapezoid rule, as an example of a numerical integration method, approximates the area under a curve as a series of trapezoids.
By summing up the areas of these trapezoids, you can estimate the value of the integral. This works well when the region to be integrated doesn’t have sharp curves or discontinuities.
For linear functions, this method becomes particularly effective because the actual area under the curve is itself a trapezoid, leading to perfect estimation.

Definite Integral

The definite integral of a function over a given interval \([x_0, x_1]\) represents the area under the curve of that function within that interval. It's calculated using calculus and provides the exact value of that area.
When approaching the integration of linear functions, the calculation of a definite integral is straightforward due to the simple nature of the function.

For a linear function \( f(x) = ax + b \), the definite integral from \( x_0 \) to \( x_1 \) can be expressed as:\[\int_{x_0}^{x_1} (ax + b) \, dx = \left[\frac{1}{2}ax^2 + bx\right]_{x_0}^{x_1}.\]When evaluated, this expression gives the exact area under the linear function from \( x_0 \) to \( x_1 \).

For linear functions, the definite integral and the trapezoid rule yield the same result, as the linear nature of the function allows the area under the curve to be represented precisely as a trapezoid.