Problem 66 - Calculus for Biology and Medicine

Question

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=x^{2}+1,-1 \leq x \leq 1\)

Step-by-Step Solution

Answer

Set up the integral \( \int_{-1}^{1} \sqrt{1 + 4x^2} \, dx \) for the curve's length.

Step 1: Identify the Arc Length Formula

The formula for the arc length of a curve given by a function \(y=f(x)\) from \(x=a\) to \(x=b\) is \[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \].

Step 2: Differentiate the Function

Given the function \(y = x^2 + 1\), find \( \frac{dy}{dx} \). The derivative \( \frac{dy}{dx} \) is \(2x\).

Step 3: Substitute into the Arc Length Formula

Substitute \( \frac{dy}{dx} = 2x\) into the arc length formula. This results in \[ L = \int_{-1}^{1} \sqrt{1 + (2x)^2} \, dx \].

Step 4: Simplify the Expression Inside the Integral

The expression inside the square root becomes \(1 + (2x)^2 = 1 + 4x^2\). Thus, the integral to be evaluated is \[ L = \int_{-1}^{1} \sqrt{1 + 4x^2} \, dx \].

Key Concepts

Understanding the Basics of Calculus

Calculus is a branch of mathematics that helps us understand changes. More specifically, calculus is split into two main categories: differential calculus and integral calculus.
Differential calculus is primarily about finding how things change. It deals with concepts like the slope of a line or the rate of change of a function. In simple terms, it's like understanding how fast something is moving or growing. Integral calculus, on the other hand, focuses on finding the accumulation of quantities, such as areas under curves.

• Calculus is essential for describing motion, growth, and areas.
• It explains how quantities evolve and accumulate over space and time.

In our problem, we use integral calculus to find the curve's length by understanding its accumulation over a range.

Exploring the Concept of Curve Length

The length of a curve, also known as its arc length, is similar to how we measure a line on a graph. For smooth curves, this requires more sophisticated methods than just using geometry. The formula for finding the arc length of a curve defined by the function \(y = f(x)\) involves integrating a specific formula.
In our exercise, we measure the length of the curve formed by the equation \(y = x^2 + 1\) between \(x = -1\) and \(x = 1\). By using calculus, we account for the curve's nature, even if it twists and turns. Unlike straight lines, curves bend, so their lengths can't be evaluated via simple addition.

• Curve length measures the total distance along a curve.
• Integral calculus helps in assessing this measurement precisely.

The Role of Integral Calculus in Finding Arc Length

Integral calculus plays a crucial role when we aim to find the length of a curve. It involves setting up an integral that measures the curve's length over the defined interval. In our given problem, we have to set up the integral using the formula:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \; dx \]
The steps invoke taking a derivative of the function and inputting it into the formula. This results in solving an integral that accounts for continuous and incremental changes along the curve.

• The formula incorporates the function's derivative, \(\frac{dy}{dx}\), capturing tiny changes along the curve.
• By integrating, we mathematically "add up" these differences to yield the total curve length.

Integral calculus helps convert these mathematical ideas into workable solutions, demonstrating its strength in real-world applications.