Question

Use the trapezoidal rule with four subdivisions to estimate \(\int_{2}^{4} x^{2} d x\).

Short answer

The approximate value of the integral is 18.625.

Step-by-step solution

Identify the interval and the number of subdivisions

The integral we want to approximate is \(\int_{2}^{4} x^2 \, dx\). The interval \([2,4]\) is the range over which we will integrate. The number of subdivisions we will use is 4, denoted as \(n = 4\).

Calculate the width of each subdivision

The width of each subdivision, denoted as \(h\), is calculated using the formula \(h = \frac{b-a}{n}\), where \(b\) is the upper limit and \(a\) is the lower limit of the integral. So here, \(h = \frac{4-2}{4} = 0.5\).

Determine the subdivision points

With a width of \(h = 0.5\), the subdivision points, starting from \(a = 2\), are calculated as follows: \(x_0 = 2\), \(x_1 = 2 + 0.5 = 2.5\), \(x_2 = 3\), \(x_3 = 3.5\), and \(x_4 = 4\).

Evaluate the function at each subdivision point

We will compute the values of \(f(x) = x^2\) at each subdivision point:- \(f(x_0) = f(2) = 4\)- \(f(x_1) = f(2.5) = 6.25\)- \(f(x_2) = f(3) = 9\)- \(f(x_3) = f(3.5) = 12.25\)- \(f(x_4) = f(4) = 16\).

Apply the trapezoidal rule formula

The trapezoidal rule formula for approximating the integral is given by:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2(f(x_1) + f(x_2) + f(x_3)) + f(x_4) \right]\]In this case, substituting the function values, we get:\[\frac{0.5}{2} \left[ 4 + 2(6.25 + 9 + 12.25) + 16 \right] = 0.25 \left[ 4 + 54.5 + 16 \right] = 0.25 \times 74.5 = 18.625\]

Final answer

The trapezoidal rule gives an estimated value for the integral \(\int_{2}^{4} x^2 \, dx\) as 18.625.

Key concepts

Numerical Integration

Numerical integration is a vital tool when it comes to approximating the value of a definite integral, especially when the integrand does not have an elementary anti-derivative or is difficult to integrate analytically. The idea is to compute an approximation by breaking down the process into simpler, manageable parts and then applying a mathematical strategy to estimate the area under the curve. For instance, one could use methods such as:

• Trapezoidal Rule
• Simpson’s Rule
• Midpoint Rule

The Trapezoidal Rule is particularly famous because it is both straightforward and often quite effective. It works by dividing the area under a curve into trapezoids and summing these to estimate the total area. Numerical integration enables us to solve physical, financial, and other real-world problems that require precise numerical solutions.

Definite Integral

The concept of a definite integral is at the heart of understanding areas and accumulations in calculus. A definite integral, denoted as \int_{a}^{b} f(x) \, dx\, represents the area under the curve of the function \(f(x)\) from the lower limit \(a\) to the upper limit \(b\). This captures the net signed area, taking into account any parts of the curve that might lie below the x-axis.

Definite integrals have numerous applications:

• Calculating areas under curves
• Determining accumulated quantities, such as distance traveled over time
• Solve problems in physics, like finding the work done by a force

The Trapezoidal Rule lets us estimate these integrals numerically by splitting the area into simpler shapes. This makes it easier to calculate when the exact answer is complex or unknown.

Subdivision

Subdivision is key when applying the Trapezoidal Rule, as it involves dividing the entire interval into smaller, uniform sections. The more subdivisions there are, represented as \(n\), the higher the potential accuracy of the approximation.

Each subdivision is determined based on its width \(h\), calculated using the formula:\[h = \frac{b-a}{n}\]where \(a\) and \(b\) are the limits of integration. After identifying \(h\), we define points \(x_0, x_1, \dots, x_n\) across the interval.

Choosing the number of subdivisions involves a tradeoff:

• More subdivisions typically enhance accuracy
• However, too many can increase computation time and complexity

Optimal subdivision balances precision and efficiency.

Calculus

Calculus is a powerful branch of mathematics focusing on change and motion, encompassing two main components: differentiation and integration.

Differentiation involves finding rates of change, like the slope of a curve at a point. Integration, on the other hand, deals with accumulation - adding up small quantities to find a total.

Integration in calculus has two main types:

• Indefinite integrals, which represent a general form of antiderivatives.
• Definite integrals, which evaluate the area under a curve over a specified interval.

Methods like the Trapezoidal Rule allow numerical integration to be performed even when the analytical approach is infeasible. With calculus playing a crucial role in sciences and engineering, techniques like this expand our ability to solve practical problems.