Question

Use the trapezoidal rule with four subdivisions to estimate \(\int_{0}^{0.8} x^{3} d x\) to four decimal places.

Short answer

0.1088

Step-by-step solution

Identify the interval and function

The integral provided is \[ \int_{0}^{0.8} x^{3} dx. \]The function is \( f(x) = x^3 \), and we need to evaluate this from \( x = 0 \) to \( x = 0.8 \).

Determine the width of each trapezoid

The interval \([0, 0.8]\) is divided into 4 equal subintervals. The width \( h \) of each subinterval is calculated as:\[ h = \frac{b - a}{n} = \frac{0.8 - 0}{4} = 0.2, \]where \( b = 0.8 \), \( a = 0 \), and \( n = 4 \).

Calculate function values at the endpoints and midpoints

Calculate \( f(x) \) at each endpoint:- \( f(0) = 0^3 = 0 \)- \( f(0.2) = (0.2)^3 = 0.008 \)- \( f(0.4) = (0.4)^3 = 0.064 \)- \( f(0.6) = (0.6)^3 = 0.216 \)- \( f(0.8) = (0.8)^3 = 0.512 \).

Apply the trapezoidal rule formula

The trapezoidal rule formula is given by:\[T = \frac{h}{2} [f(x_0) + 2(f(x_1) + f(x_2) + f(x_3)) + f(x_4)],\]where \( x_0 = 0 \), \( x_1 = 0.2 \), \( x_2 = 0.4 \), \( x_3 = 0.6 \), \( x_4 = 0.8 \).

Substitute values into the trapezoidal rule

Substitute the known function values into the formula:\[T = \frac{0.2}{2} [0 + 2(0.008 + 0.064 + 0.216) + 0.512].\]Calculate inside the brackets first:\[2(0.008 + 0.064 + 0.216) = 2(0.288) = 0.576.\]Thus,\[T = 0.1 [0 + 0.576 + 0.512] = 0.1 [1.088] = 0.1088.\]

Round to four decimal places

The calculated value of the integral using the trapezoidal rule with four decimal places is 0.1088.

Key concepts

Numerical Integration

Numerical integration is a powerful tool used in calculus to estimate the value of definite integrals, especially when an exact analytical solution is difficult or impossible to find. Instead of finding the antiderivative of a function, numerical methods break the problem into smaller parts and approximate the area under the curve.
One of the most widely used techniques is the trapezoidal rule. This method approximates the region under a curve by dividing it into a series of trapezoids. By calculating the areas of these trapezoids and summing them up, we obtain an estimated value for the integral.
Can't integrate complex functions easily? Here are some advantages of using numerical integration:

• It handles complex, awkward functions where traditional integration fails or is tough.
• Flexibility allows for varying degrees of accuracy, depending on how much you subdivide the interval.
• It’s a foundation for more advanced methods, like Simpson's rule and Gaussian quadrature, which offer even more precision.

The trapezoidal rule is straightforward and serves as an introductory concept to the world of numerical approximations for integrals.

Approximation Methods

Approximation methods, like the trapezoidal rule, are essential in mathematics for situations where exact answers are impractical. These methods transform difficult integrals into manageable calculations. They provide a way to achieve solutions by balancing speed and accuracy.
The trapezoidal rule involves a few simple steps:

• Divide the integration interval into equal sub-intervals (in this exercise, the interval is from 0 to 0.8).
• Calculate the function value at each endpoint and use these values to form trapezoids under the curve.
• Apply the trapezoidal formula to estimate the integral by summing the areas of the trapezoids.

This incremental approach might not give exact values, but it efficiently provides a very close estimate. By increasing the number of subdivisions, the accuracy of these approximation methods generally improves. The trapezoidal rule strikes a balance by utilizing simple calculations while achieving good accuracy for many practical applications.

Calculus Problem-Solving

In calculus, problem-solving often requires innovative methods to deal with complex equations and integrations. The trapezoidal rule is one such method that simplifies the process. It changes how we look at integrals by transforming them from theoretical exercises into solvable numerical problems.
For instance, in the original exercise, the trapezoidal rule was used to estimate the integral from 0 to 0.8 of the function \( x^3 \). By following systematic steps, such as dividing the interval and calculating function values, we built a practical solution.
The key to effective calculus problem-solving with these methods includes understanding the underlying mathematical principles and knowing when and how to apply them. Here are a few tips to keep in mind:

• Always verify if the function is continuous and defined within the interval before using numerical methods.
• Consider increasing the number of sub-intervals to obtain more accurate results.
• Cross-check results with alternative methods when possible for verification.

Using numerical methods like the trapezoidal rule not only hones calculus skills but also enhances problem-solving abilities by providing practical tools to tackle real-world challenges.