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.
![**Calculus: Finding the Area Between Curves**
In this exercise, we aim to find the area of the region enclosed between two curves, specifically the curves represented by the equations:
1. \( 4x + y^2 = 12 \)
2. \( x = y \)
### Steps to Find the Area
1. **Set the Equations Equal to Find Points of Intersection:**
To find the points where these curves intersect, set \( x = y \) into the first equation:
\[
4x + y^2 = 12 \implies 4y + y^2 = 12 \implies y^2 + 4y - 12 = 0
\]
Solve this quadratic equation for \( y \):
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{16 + 48}}{2} = \frac{-4 \pm 8}{2}
\]
This gives us two solutions for \( y \):
\[
y = 2 \quad \text{and} \quad y = -6
\]
Since \( x = y \), the points of intersection are \( (2, 2) \) and \( (-6, -6) \).
2. **Set Up the Integral:**
The area between the curves can be found using the integral:
\[
\text{Area} = \int_{c}^{d} \left[ f(y) - g(y) \right] dy
\]
Here, \( f(y) \) is the rightmost function (in this case, \( x = \frac{12 - y^2}{4} \) solved from \( 4x + y^2 = 12 \)) and \( g(y) \) is the leftmost function (in this case, \( x = y \)).
3. **Boundaries of Integration:**
We integrate with respect to \( y \) from \( -6 \) to \( 2 \):
\[
\text{Area} = \int_{-6}^{2} \left( \frac{12 - y^2}{4} - y \right) dy
\]
4. **Simplify the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa090e79e-b63b-42aa-a337-95657eeca289%2F9d4de1b4-e097-454b-844b-f5ad69d95c62%2F44vj5aj_processed.jpeg&w=3840&q=75)
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