Problem 171 - Calculus Volume 1

Question

For the following exercises, find the lengths of the functions of \(x\) over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it. $$y=x^{3 / 2} \text { from }(0,0) \text { to }(1,1)$$

Step-by-Step Solution

Answer

The arc length is approximately 1.4789 units.

Step 1: Identify the formula for arc length

The formula for the arc length of a function \( y = f(x) \) over the interval \([a, b]\) is given by \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \). We will use this formula to find the length of the function \( y = x^{3/2} \) from \((0,0)\) to \((1,1)\).

Step 2: Compute the derivative \( f'(x) \)

First, find the derivative of the function \( y = x^{3/2} \). Using the power rule, the derivative is \( f'(x) = \frac{3}{2}x^{1/2} \).

Step 3: Substitute \( f'(x) \) into arc length formula

Substitute \( f'(x) = \frac{3}{2}x^{1/2} \) into the arc length formula. So, the integrand becomes \( \sqrt{1 + \left(\frac{3}{2}x^{1/2}\right)^2} = \sqrt{1 + \frac{9}{4}x} \).

Step 4: Set up the integral for arc length

Set up the integral for the arc length over the interval \([0, 1]\): \( L = \int_0^1 \sqrt{1 + \frac{9}{4}x} \, dx \).

Step 5: Evaluate the integral

Evaluating this integral by hand can be complex, so it's best to use technology. Using a numerical method, such as with a calculator or software, we find that \( \int_0^1 \sqrt{1 + \frac{9}{4}x} \, dx \approx 1.4789 \).

Step 6: State the arc length

The approximate length of the function \( y = x^{3/2} \) from the point \((0,0)\) to the point \((1,1)\) is approximately 1.4789 units.

Key Concepts

Derivative of a Function

Calculus often begins with understanding the **derivative of a function**. The derivative represents how a function changes at any point. It's like calculating the speed of a moving car—how fast the position changes over time. In our exercise, the function given is \( y = x^{3/2} \). To find the derivative, we apply rules of differentiation, such as the **Power Rule in Calculus**. By using this rule, we find that the derivative of \( x^{3/2} \) is \( f'(x) = \frac{3}{2}x^{1/2} \). This means that at any point \( x \), the function's rate of change is \( \frac{3}{2}x^{1/2} \). Such derivatives play a crucial role in determining arc length later.

Integral Evaluation

Once we have the derivative, we move to **integral evaluation** to find arc length. Mathematically, an integral sums up small slices of data to provide complete information—in this case, the total length of the curves between two points. We use the formula:\[L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\]Here, \( f'(x) \) is our derivative, \( \frac{3}{2}x^{1/2} \). Plug this into the formula to get the integral you need to evaluate. Unfortunately, not all integrals can be solved by hand. This is why technology or calculators are often used.

Numerical Approximation

In cases where calculus reaches complex integrals, like ours, we can use **numerical approximation**. Computers and calculators handle these with ease. **Numerical methods** such as Simpson's Rule or Trapezoidal Rule provide approximations when basic calculus falls short. For the integral \( \int_0^1 \sqrt{1 + \frac{9}{4}x} \, dx \), a numerical approach shows the arc length is about 1.4789 units. Remember, these approximations are useful and generally very accurate, especially with technology's assistance.

Power Rule in Calculus

In calculus, the **Power Rule** simplifies the process of finding derivatives of polynomial functions. If you have a function \( y = x^n \), its derivative is \( f'(x) = nx^{n-1} \). This process reduces complex derivations to straightforward steps. In our case with \( y = x^{3/2} \), apply the Power Rule:- Multiply the power by the term: \( 3/2 \times x^{3/2 - 1} \).- Simplify to get \( \frac{3}{2}x^{1/2} \).Such rules save you time and ensure precision. Keep practicing these rules with various functions to gain mastery.