Problem 2 - Calculus for Biology and Medicine

Question

Use the midpoint rule to approximate each integral with the specified value of \(n\). $$ \int_{-1}^{0}(x+1)^{2} d x, n=5 $$

Step-by-Step Solution

Answer

The approximate value of the integral is 0.33.

Step 1: Determine the width of each subinterval

The interval from -1 to 0 has a width of 1. We need to divide this interval into 5 equal parts since \( n=5 \). The width \( \Delta x \) of each subinterval is thus \( \Delta x = \frac{(0 - (-1))}{5} = \frac{1}{5} = 0.2 \).

Step 2: Find the midpoints of each subinterval

The midpoints for the subintervals are calculated by finding the middle point between the endpoints of each subinterval. The subintervals are: 1. \([-1, -0.8]\), midpoint: \(-0.9\)2. \([-0.8, -0.6]\), midpoint: \(-0.7\)3. \([-0.6, -0.4]\), midpoint: \(-0.5\)4. \([-0.4, -0.2]\), midpoint: \(-0.3\)5. \([-0.2, 0]\), midpoint: \(-0.1\)

Step 3: Evaluate the function at each midpoint

Substitute each midpoint into the function \((x+1)^2\):1. For \(x = -0.9\), \((-0.9+1)^2 = 0.1^2 = 0.01\)2. For \(x = -0.7\), \((-0.7+1)^2 = 0.3^2 = 0.09\)3. For \(x = -0.5\), \((-0.5+1)^2 = 0.5^2 = 0.25\)4. For \(x = -0.3\), \((-0.3+1)^2 = 0.7^2 = 0.49\)5. For \(x = -0.1\), \((-0.1+1)^2 = 0.9^2 = 0.81\)

Step 4: Calculate the approximate integral using the midpoint rule

The approximate integral \( I \) using the midpoint rule is the sum of the function values at the midpoints, each multiplied by \( \Delta x \):\[ I = 0.2(0.01 + 0.09 + 0.25 + 0.49 + 0.81) \]First, compute the sum inside the parentheses:\[ 0.01 + 0.09 + 0.25 + 0.49 + 0.81 = 1.65 \]Then multiply by \( \Delta x = 0.2 \):\[ I = 0.2 \times 1.65 = 0.33 \]

Step 5: Interpret the result

The value computed using the midpoint rule gives us an approximation of the integral. In this case, \( \int_{-1}^{0}(x+1)^{2} \, dx \approx 0.33 \) with \( n=5 \).

Key Concepts

Integral Approximation

The concept of integral approximation is a fundamental idea in calculus, where we approximate the exact value of a definite integral using numerical methods. Instead of finding the exact area under a curve analytically, integral approximation involves breaking down the area into simpler, calculated sections that are easy to compute. This is especially helpful when dealing with functions that are difficult or impossible to integrate analytically.

For our exercise, we're tasked with finding the approximate value of the integral of \((x+1)^2\) from \(-1\) to \(0\), using \(n=5\) subintervals. By employing the midpoint rule, we can estimate the integral's value effectively. This numerical technique helps us find a practical solution where algebraic methods fall short.

Subinterval Calculation

Dividing the interval into subintervals is an essential step when using numerical methods like the midpoint rule. For our given interval of \([-1, 0]\), we must first calculate how wide each subinterval will be. This is done by subtracting the lower bound of the interval from the upper bound and then dividing by the number of subintervals \(n\). Here, the interval width is \(1\) (from \(-1\) to \(0\)), and with \(n=5\), each subinterval's width (94x) is \(rac{1}{5} = 0.2\).

Knowing the width of each subinterval allows us to partition the interval into smaller sections, each of which will contribute equally to the final integral approximation. The process also allows us to find midpoints within each subinterval, which are crucial for applying the midpoint rule effectively.

Function Evaluation

Once we have calculated the subintervals, the next step in the process is evaluating the function at specific points. In the midpoint rule, we use the midpoint of each subinterval to evaluate the function.

For our function \((x+1)^2\), we calculate function values at midpoints. The midpoints are simple averages of the subinterval endpoints:

• 91e.g., for \([-1, -0.8]\), the midpoint is \(-0.9\) and \((-0.9+1)^2 = 0.01\).
• 91Similarly, calculate for other midpoints: \(-0.7, -0.5, -0.3,\) and \(-0.1\).

By evaluating the function at these midpoints, we gather all necessary function values needed for the next step.

Numerical Integration

Numerical integration involves combining the calculated function values at each midpoint with the width of the subintervals to approximate the definite integral.

Following the midpoint rule, calculate the overall approximate integral by multiplying the sum of the function values with the subinterval width (94x):

We first add all function values obtained from the midpoints: \[(0.01 + 0.09 + 0.25 + 0.49 + 0.81) = 1.65\]
Multiply this sum by the width \(94x = 0.2\): \(I = 0.2 imes 1.65 = 0.33\)

This provides us with the approximate value of the integral, demonstrating that numerical integration can be a powerful tool in estimating areas under curves when traditional methods are complex or unfeasible.