Problem 11 - Calculus for Biology and Medicine

Question

Approximate $$ \int_{-1}^{1}\left(2+x^{2}\right) d x $$ using five equal subintervals.

Step-by-Step Solution

Answer

The approximate value of the integral using five subintervals is 4.64.

Step 1: Define the Integral and Subintervals

The integral given is \( \int_{-1}^{1}(2 + x^2) \, dx \). We need to approximate it using five equal subintervals between -1 and 1. The interval \([-1, 1]\) can be divided into five equal parts of length \( \Delta x = \frac{1 - (-1)}{5} = \frac{2}{5} = 0.4 \). The endpoints of the subintervals are \(-1, -0.6, -0.2, 0.2, 0.6, 1\).

Step 2: Determine Midpoints of Subintervals

To use the midpoint rule for approximation, calculate the midpoints of each subinterval:- For \([-1, -0.6]\), midpoint is \(-0.8\).- For \([-0.6, -0.2]\), midpoint is \(-0.4\).- For \([-0.2, 0.2]\), midpoint is \(0\).- For \([0.2, 0.6]\), midpoint is \(0.4\).- For \([0.6, 1]\), midpoint is \(0.8\).

Step 3: Evaluate Function at Midpoints

Evaluate the function \( f(x) = 2 + x^2 \) at each midpoint:- \( f(-0.8) = 2 + (-0.8)^2 = 2 + 0.64 = 2.64 \)- \( f(-0.4) = 2 + (-0.4)^2 = 2 + 0.16 = 2.16 \)- \( f(0) = 2 + 0^2 = 2 \)- \( f(0.4) = 2 + 0.4^2 = 2 + 0.16 = 2.16 \)- \( f(0.8) = 2 + 0.8^2 = 2 + 0.64 = 2.64 \)

Step 4: Apply Midpoint Rule

Use the midpoint rule to approximate the integral: \[ \int_{-1}^{1}(2 + x^2) \, dx \approx \Delta x \sum_{i=1}^{5} f(x_i^*) \] where \( \Delta x = 0.4 \) and \( x_i^* \) are the midpoints calculated.Compute the expression:\[ 0.4 \times (2.64 + 2.16 + 2 + 2.16 + 2.64) = 0.4 \times 11.6 = 4.64 \]

Key Concepts

Numerical Integration

Numerical integration is a technique used to calculate the integral of a function when finding an exact solution is difficult or impossible. It involves approximating the area under a curve by breaking it down into simpler, easily computable shapes, such as rectangles or trapezoids.
Numerical integration is especially useful when dealing with complex functions that do not have a straightforward antiderivative, like most real-world functions.
By using numerical integration, we can still estimate the area under the curve and thus approximate the integral.

• It is fundamental in many scientific and engineering applications.
• Allows approximation of definite integrals with specified bounds.
• Useful when dealing with data points rather than continuous functions.

Numerical methods such as the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule are popular techniques for this purpose.

Approximation Techniques

Approximation techniques involve methods to estimate a mathematical quantity with a simpler, more manageable value. They play a key role in mathematics, particularly in cases where exact answers are difficult to obtain.
One common approximation technique is using the Midpoint Rule to estimate the integral of a function over a certain interval.
This involves taking the midpoint of each subinterval, evaluating the function at these points, and summing up the results to get an overall estimate of the integral.

• Approximation makes complex problems solvable or more manageable.
• Techniques can differ in accuracy and complexity.
• They are crucial in computational mathematics.

In the Midpoint Rule, after dividing the integration interval into smaller subintervals, the function values at midpoints provide a good approximation of the actual integral.

Integral Calculus

Integral calculus is a branch of calculus focused on finding the integral of functions. It seeks to determine the total accumulation of quantities, such as areas under curves, lengths, and volumes.
Integrals calculate the net area between a curve and the x-axis, providing measures of total change when considering functions over intervals.
Integral calculus has two main components: definite and indefinite integrals.

• Definite integrals compute the accumulation over a specific interval.
• Indefinite integrals represent a family of functions and their antiderivatives.
• Integral calculus is extensively applied in physics, engineering, and economics to model various phenomena.

In practical scenarios, especially in definite integrals, numerical solutions like the Midpoint Rule come in handy when functions are difficult to integrate analytically.