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.
![### Curve Length Calculation
#### Problem Statement
Find the length of the curve defined by the parametric equations:
\[ x = 5 \sin t - 5t \cos t, \quad y = 5 \cos t + 5t \sin t, \quad 0 \leq t \leq \frac{\pi}{2} \]
#### Objective
Determine the length of the curve within the given interval.
The formula for the length \(L\) of a curve defined by parametric equations \( x = f(t) \) and \( y = g(t) \) over an interval \([a, b]\) is given by:
\[ L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \]
#### Solution Steps
1. **Calculate the derivatives:**
- Find \( \frac{dx}{dt} \)
- Find \( \frac{dy}{dt} \)
2. **Substitute the derivatives into the length formula:**
3. **Evaluate the integral:**
#### Given Data:
\[ x = 5 \sin t - 5 t \cos t \]
\[ y = 5 \cos t + 5 t \sin t \]
\[ 0 \leq t \leq \frac{\pi}{2} \]
#### Length Calculation
\[ \text{The length is} \, \boxed{\text{units}}. \]
*(Type an exact answer, using \(\pi\) as needed.)*
This transcribed and described the educational task of finding the length of a parametric curve within a specified interval, detailing the necessary formulas and steps involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59970047-9688-41c7-9954-29dec40053a6%2F0bf3e23d-9ebf-4fb8-b59f-3cc722617080%2Fn7om2hp_processed.png&w=3840&q=75)
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