Problem 35 - Calculus for Biology and Medicine

Question

Approximate $$\int_{-2}^{2}\left(2+x^{2}\right) d x$$ using four equal subintervals and left endpoints.

Step-by-Step Solution

Answer

The approximate value of the integral is 14.

Step 1: Determine the Width of Each Subinterval

The integral covers the range from \(-2\) to \(2\). Total width of the interval is \(2 - (-2) = 4\) units. With four equal subintervals, the width of each subinterval \(\Delta x\) is \(\frac{4}{4} = 1\).

Step 2: Identify the Left Endpoints

Since we are using left endpoints, identify them as \(-2\), \(-1\), \(0\), and \(1\) for each of the four subintervals.

Step 3: Evaluate the Function at Each Left Endpoint

The function to evaluate is \(f(x) = 2 + x^2\). Calculate the function values at each left endpoint:- \(f(-2) = 2 + (-2)^2 = 2 + 4 = 6\)- \(f(-1) = 2 + (-1)^2 = 2 + 1 = 3\)- \(f(0) = 2 + (0)^2 = 2\)- \(f(1) = 2 + (1)^2 = 2 + 1 = 3\)

Step 4: Apply the Left Endpoint Riemann Sum Formula

Use the formula for the left endpoint Riemann sum: \[L = \Delta x \sum_{i=1}^{n} f(x_i) \]Substitute the values calculated: \[L = 1 \times (f(-2) + f(-1) + f(0) + f(1)) = 1 \times (6 + 3 + 2 + 3) = 14\]

Step 5: Present the Approximation

The approximate value of the integral \(\int_{-2}^{2} (2 + x^2) \, dx\) using four left endpoints and four subintervals is \(14\).

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval on the x-axis. In mathematical terms, the definite integral of a function \( f(x) \) from \( a \) to \( b \) is written as \( \int_{a}^{b} f(x) \, dx \). This integral provides a numeric value that gives insights into the overall behavior of the function on that interval.

The definite integral is closely tied to the concept of area. If \( f(x) \) is above the x-axis between \( a \) and \( b \), the integral equals the area under \( f(x) \) and above the x-axis. When dealing with curves below the x-axis, those areas are considered negative. The final result of a definite integral might be positive, negative, or zero, depending on the function's position relative to the x-axis.

Definite integrals also have applications in various fields such as physics and engineering, where they help in calculating things like displacement, net change, and accumulated quantities. Understanding definite integrals is fundamental to analyzing physical systems and mathematical models.

Approximation Methods

Sometimes, finding the exact value of a definite integral can be challenging or impossible using basic calculus techniques. In such cases, approximation methods like Riemann sums come in handy. These methods break down the integral's overall problem into smaller, manageable pieces.

One popular approximation method is the Riemann sum, which divides the interval of integration into a certain number of subintervals. By calculating the function's value at specific points within these subintervals, we can approximate the total area under the curve. There are different types of Riemann sums depending on the points used:

• **Left Endpoint Riemann Sum:** Uses the left endpoint of each subinterval for evaluation.
• **Right Endpoint Riemann Sum:** Uses the right endpoint of each subinterval.
• **Midpoint Riemann Sum:** Uses the midpoint of each subinterval.

These methods provide estimates that improve in accuracy as more subintervals are used, leading to a more precise approximation of the definite integral's true value.

Subintervals

Subintervals are the smaller sections into which the overall interval of integration is divided when using approximation methods like Riemann sums. The process of dividing the original interval helps simplify the computation of the integral's value by focusing on smaller, more manageable pieces.

To create subintervals, we begin by determining the width of each section, often denoted as \( \Delta x \). This width is calculated as the total length of the interval divided by the number of subintervals. For example, in the exercise given, the interval from \(-2\) to \(2\) was divided into four equal parts, resulting in a width \( \Delta x = 1 \).

Each subinterval contains a specific point, such as the left endpoint, right endpoint, or midpoint, which is used to evaluate the function's value in approximation methods. The choice of points impacts the Riemann sum's accuracy. Greater numbers of subintervals generally lead to more accurate approximations, as they allow for a finer integration of the function's behavior across the entire interval.