Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![### Calculating the Arc Length of a Curve
To determine the arc length of the curve described by the function:
\[ y = \frac{x^4}{4} + \frac{1}{8x^2} \]
over the interval \([1,4]\), follow these steps:
#### Step-by-Step Process
1. **Determine the Derivative of \( y \)**:
- The function given is \( y = \frac{x^4}{4} + \frac{1}{8x^2} \).
- Compute the derivative \( \frac{dy}{dx} \).
2. **Arc Length Formula**:
- Use the arc length formula for a function \( y = f(x) \) from \( a \) to \( b \):
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
3. **Integrate**:
- Plug in the derivative \( \frac{dy}{dx} \) into the formula.
- Calculate the definite integral from \( x = 1 \) to \( x = 4 \).
#### Key Points to Remember
- **Derivative Calculation**: Ensure the correct computation of the derivative.
- **Substituting in Formula**: Accurately substitute the derivative into the arc length formula.
- **Integration Limits**: Use the correct limits of integration which are \( [1, 4] \).
#### Visualization
1. **Curve**:
- The curve \( y = \frac{x^4}{4} + \frac{1}{8x^2} \) is a combination of polynomial and rational functions.
2. **Interval**:
- The interval [1,4] specifies the part of the curve for which we are finding the arc length.
#### Final Step
After finding the exact arc length through integration, you can fill in the answer in the provided box:
\[ \text{The length of the curve is } \boxed{ \ } \]
**Note**: Be sure to provide the answer in its exact form, using radicals as needed.
Would you like to go through the detailed steps of computing the derivative and the integral, or is there another specific part of the problem you need assistance with?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03b209f4-7a0d-487f-8990-c972f466589b%2F1f023aae-0f10-4ebf-b23b-31b42fad6f3e%2Fzx1eb.jpeg&w=3840&q=75)
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