Problem 10 - Calculus for Biology and Medicine

Question

Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) \(\int_{-1}^{0} x^{3} d x, n=5\)

Step-by-Step Solution

Answer

The approximate value of the integral is \(-0.26\).

Step 1: Understand the Trapezoidal Rule Formula

The trapezoidal rule is used to approximate the definite integral of a function. For a function \(f(x)\) integrated from \(a\) to \(b\) with \(n\) equally spaced subintervals, the trapezoidal rule formula is: \[\int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left( f(a) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(b) \right)\]where \(x_i = a + i\frac{b-a}{n}\) for \(i = 1, 2, \ldots, n-1\).

Step 2: Define the Function and Interval

Given the integral \(\int_{-1}^{0} x^{3} \, dx\) with \(n=5\), we have:- \(f(x) = x^3\)- Interval \([-1, 0]\)- \(a = -1, b = 0\)

Step 3: Calculate Subinterval Width

The width of each subinterval \(h\) is given by:\[h = \frac{b-a}{n} = \frac{0 - (-1)}{5} = \frac{1}{5} = 0.2\]

Step 4: Determine the Subinterval Points

Calculate the points \(x_i\) where the function will be evaluated:- \( x_0 = -1.0 \)- \( x_1 = x_0 + 0.2 = -0.8 \)- \( x_2 = x_1 + 0.2 = -0.6 \)- \( x_3 = x_2 + 0.2 = -0.4 \)- \( x_4 = x_3 + 0.2 = -0.2 \)- \( x_5 = x_4 + 0.2 = 0.0 \)

Step 5: Evaluate the Function at Given Points

Calculate \(f(x_i)\) for each subinterval point:- \(f(-1.0) = (-1.0)^3 = -1.0\)- \(f(-0.8) = (-0.8)^3 = -0.512\)- \(f(-0.6) = (-0.6)^3 = -0.216\)- \(f(-0.4) = (-0.4)^3 = -0.064\)- \(f(-0.2) = (-0.2)^3 = -0.008\)- \(f(0.0) = (0.0)^3 = 0.0\)

Step 6: Substitute Values into Trapezoidal Rule Formula

Using the trapezoidal rule formula, substitute the values:\[\int_{-1}^{0} x^{3} \, dx \approx \frac{0 - (-1)}{2 \times 5} \left( -1.0 + 2(-0.512) + 2(-0.216) + 2(-0.064) + 2(-0.008) + 0.0 \right)\]Simplify:\[\approx 0.1 \left( -1.0 + (-1.024) + (-0.432) + (-0.128) + (-0.016) \right)\]\[\approx 0.1 (-2.6) = -0.26\]

Step 7: Final Answer

The trapezoidal rule approximation of the integral \(\int_{-1}^{0} x^3 \, dx\) with \(n=5\) is \(-0.26\).

Key Concepts

Numerical Integration

Often, we need to find the area under a curve when exact analytical solutions are hard to obtain. That's where numerical integration shines. It provides methods to approximate definite integrals of various functions, making complex calculus problems more manageable.

Numerical integration is key when working with empirical data, difficult functions, or functions without simple antiderivatives. Common rules used for approximations include:

• Trapezoidal Rule
• Simpson's Rule
• Midpoint Rule

Each rule varies in complexity and accuracy. The choice depends on the function and the desired precision. In our problem, the trapezoidal rule was used, which uses the concept of fitting trapezoids under a curve to estimate the area.

Definite Integral

A definite integral computes the accumulated value, like area, between a function and the x-axis over a given interval. The definite integral of a function from point \(a\) to point \(b\) is denoted:\[\int_{a}^{b} f(x) \, dx\]This integral represents the total area under the curve \(f(x)\) between \(x = a\) and \(x = b\).

Unlike indefinite integrals, which require an arbitrary constant, definite integrals produce a number, representing this total accumulated value. They're essential for understanding the net effects of changing quantities, such as total income over time based on a variable rate.

For our exercise, the definite integral of \(x^3\) from -1 to 0 was calculated using the trapezoidal rule, which approximates the integral as \(-0.26\).

Polynomial Function

Polynomial functions are expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial can be written as:\[P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]These functions are fundamental in mathematics due to their simple structure and the variety of problems they can model.

In the given exercise, the function \(f(x) = x^3\) is a simple cubic polynomial. Cubic polynomials can model certain types of growth and decay, among other phenomena, and understanding their behavior is crucial in fields like physics and engineering.

By analyzing polynomial functions this way, one can predict trends, estimate values, and understand complex relationships in a more manageable manner.