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.
![**Problem Statement:**
Find the total area bounded by the curves \( f(x) = \sin(2x) \) and \( y = \cos(x) \) in the interval \(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\). Also, graph the curves and shade the area.
**Graphical Representation:**
1. **Graph Plotting:**
- Plot the curve \( f(x) = \sin(2x) \).
- Plot the curve \( y = \cos(x) \).
2. **Shading the Area:**
- Identify the points of intersection between the curves within the given interval \(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\).
- Shade the region bounded by the curves between these points of intersection, representing the area to be calculated.
**Solution Steps:**
1. **Intersection Points:**
- Set \( \sin(2x) = \cos(x) \) and solve for \( x \) in the interval \(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\).
2. **Definite Integrals:**
- Use integration to calculate the area between the curves from the left intersection point to the right intersection point.
- Split the integral if the points of intersection change the function that is on top within the interval.
3. **Total Area Calculation:**
- Sum the absolute values of the integrals to find the total bounded area.
This will be a detailed step-by-step process that might include analytical and numerical methods to find the points and areas. A graph will visually demonstrate the curves and the shaded bounded area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39f80a36-5866-477b-b28d-154d74fc6779%2F2b79a325-432c-47d6-a1b3-e77859e954d8%2Fohprw5f_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 3 images
