Question

Find the exact value of \(\int_{2}^{4} x^{2} d x .\) Find the error of approximation between the exact value and the value calculated using the trapezoidal rule with four subdivisions. Draw a graph to illustrate.

Short answer

The exact integral is \( \frac{56}{3} \), trapezoidal approximation is 18.75, and error is \( \frac{1}{12} \).

Step-by-step solution

Find the Antiderivative

To find the exact value of the definite integral, we first determine the antiderivative of the function. For the function \( x^2 \), the antiderivative is \( \frac{x^3}{3} + C \), where \( C \) is the constant of integration.

Evaluate the Definite Integral

Now, we need to evaluate this antiderivative from the limits of 2 to 4. So we compute this as \( \left[ \frac{x^3}{3} \right]_2^4 = \frac{4^3}{3} - \frac{2^3}{3} = \frac{64}{3} - \frac{8}{3} = \frac{56}{3} \).

Trapezoidal Rule Setup

The trapezoidal rule with \( n = 4 \) subdivisions divides the interval \([2, 4]\) into 4 equal parts. The subinterval length \( \Delta x \) is \( \frac{4-2}{4} = 0.5 \). The x-values at each partition are 2, 2.5, 3, 3.5, and 4.

Calculate Function Values

Evaluate the function \( f(x) = x^2 \) at each partition point: \( f(2) = 4 \), \( f(2.5) = 6.25 \), \( f(3) = 9 \), \( f(3.5) = 12.25 \), and \( f(4) = 16 \).

Apply the Trapezoidal Rule

Using the trapezoidal rule formula:\[\text{Approximation} = \frac{\Delta x}{2} \times \left( f(2) + 2f(2.5) + 2f(3) + 2f(3.5) + f(4) \right)\]This gives:\[\frac{0.5}{2} \times (4 + 2 \times 6.25 + 2 \times 9 + 2 \times 12.25 + 16) = 0.25 \times (4 + 12.5 + 18 + 24.5 + 16) = 0.25 \times 75 = 18.75\]

Error of Approximation

The error of approximation is the absolute difference between the exact value and the trapezoidal approximation: \( \left| \frac{56}{3} - 18.75 \right| = \left| \frac{56}{3} - \frac{75}{4} \right| = \left| \frac{224}{12} - \frac{225}{12} \right| = \frac{1}{12} \).

Graph Illustration

To illustrate, plot the function \( y = x^2 \) from \( x = 2 \) to \( x = 4 \) and show the trapezoids formed by each subdivision. Shade the area under the curve above the x-axis, which the exact integral evaluates, and compare with the area of the trapezoids for visual understanding.

Key concepts

Trapezoidal Rule

The trapezoidal rule is an essential numerical method used in integral calculus to approximate a definite integral. Instead of calculating the exact area under a curve, which can be complex, especially if we don't have an antiderivative, we approximate it using simpler geometric shapes—trapezoids. The method involves dividing the integration interval into smaller subintervals.

• This subdivision breaks the curve into a series of straight lines connecting function values at these points.
• In our specific exercise, the interval \[ [2, 4] \] is divided into 4 parts, resulting in subintervals of length \[ \Delta x = 0.5 \].

The trapezoidal rule formula helps us calculate the total area of these trapezoids:
\[ \text{Approximation} = \frac{\Delta x}{2} \times \left( f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right) \] This approach offers a practical approximation, trading precision for computational simplicity.

Definite Integral

The definite integral represents the exact area under a curve from one point to another along the x-axis. Unlike an indefinite integral, which includes a constant of integration, a definite integral provides a specific value.
In our example problem \[ \int_{2}^{4} x^{2} dx \],

• We integrate the function \[ x^2 \] from \[ x = 2 \] to \[ x = 4 \].
• This process involves calculating the antiderivative, evaluating it at the upper limit, then subtracting its value at the lower limit.

This gives us the exact integral value from 2 to 4 as \[ \frac{56}{3} \], which precisely measures the area under the curve \[ y = x^2 \] between these points.

Error Approximation

In numerical methods like the trapezoidal rule, error approximation is a critical concept. It measures how close or far the numerical approximation is from the exact integral value. Identifying this error helps assess the effectiveness of the trapezoidal rule. We calculate it by finding the absolute difference between the exact value and the approximation:
\[ \text{Error} = \left| \text{Exact Value} - \text{Approximation} \right| \]

• For the integral \([2, 4] x^2 \), the exact value is \[ \frac{56}{3} \].
• The trapezoidal approximation calculates to 18.75.

Thus, the approximation error is \[ \frac{1}{12} \], indicating that the trapezoidal rule is not perfect but provides a close approach, depending on the number of subdivisions used and the function's nature.

Antiderivative

Antiderivatives play a foundational role in integral calculus, especially when calculating definite integrals. An antiderivative of a function is another function that, when differentiated, returns the original function. It effectively works in the opposite direction of taking a derivative.

• For example, the antiderivative of \({ x^2 } \) is \({ \frac{x^3}{3} + C } \), where \( C \) represents the constant of integration.

When solving definite integrals, we utilize the antiderivative by evaluating it over the given interval to calculate the precise area under the curve. This method involves the Fundamental Theorem of Calculus, which states that if \( F(x) \) is the antiderivative of \( f(x) \), then the definite integral from \( a \) to \( b \) is \[ F(b) - F(a) \]. This provides the exact answer for an integral, such as our example \({ \int_{2}^{4} x^{2} dx = \frac{56}{3} } \).