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 amount of bounded area between \( y = \sin x \) and the line \( y = -1 + \frac{1}{\pi} x \) from \( x = 0 \) to \( x = \pi \).
**Solution Explanation:**
First, we need to find the points of intersection of the curves \( y = \sin x \) and \( y = -1 + \frac{1}{\pi} x \). Solving for \( x \) when \( \sin x = -1 + \frac{1}{\pi} x \):
1. Set \( \sin x = -1 + \frac{1}{\pi} x \)
2. Determine solutions within the interval \([0, \pi]\)
Next, we would set up the integral to compute the area between the curves. This involves integrating the difference between the top function and the bottom function over the interval of interest. Assuming \( \sin x \) is above the line \( -1 + \frac{1}{\pi} x \) throughout the interval:
\[ \text{Area} = \int_0^\pi \left( \sin x - \left(-1 + \frac{1}{\pi} x\right) \right) \, dx \]
Simplify and integrate this expression to find the enclosed area:
\[ \text{Area} = \int_0^\pi \left( \sin x + 1 - \frac{1}{\pi} x \right) \, dx \]
Evaluate this integral to find the exact area.
**Diagrams:**
1. Draw the sine curve \( y = \sin x \) between \( x = 0 \) to \( x = \pi \).
2. Draw the line \( y = -1 + \frac{1}{\pi} x \) within the same interval.
3. Shade the area between these two curves, which represents the region whose area we are calculating.
Such graphical representations often help in better understanding the intersection points and the bounded area to be calculated.
**Note:** For a step-by-step solution involving exact numerical integration or graph plotting, a mathematical computation software or a graphing calculator may be employed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F672bf286-8abe-4b07-9ca1-0d5b2612956c%2Fc6084f54-0d09-42a0-b3c1-c9d6d3291f97%2F4yfaovm_processed.jpeg&w=3840&q=75)
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