B) For curve given in polar coordinates by r = 1 + cos0: i) Graph it i) Set-up an integral for the area that the curve encloses. Don't evaluate the integral
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.
![**Understanding Polar Coordinates and Area Calculation:**
### Problem Statement
**B)** For the curve given in polar coordinates by \( r = 1 + \cos(\theta) \):
1. **Graph it.**
2. **Set up an integral for the area that the curve encloses. Do not evaluate the integral.**
**Graph:**
The blank graph provided is a Cartesian plane grid with a noticeable emphasis on the central axes. Use this grid to plot the given polar equation \( r = 1 + \cos(\theta) \).
**Steps to Graph the Polar Curve:**
1. Plot points for values of \(\theta\) from \(0\) to \(2\pi\) radians (360 degrees).
2. For \(\theta = 0\), \(r = 1 + \cos(0) = 2\).
3. For \(\theta = \pi/2\), \(r = 1 + \cos(\pi/2) = 1\).
4. For \(\theta = \pi\), \(r = 1 + \cos(\pi) = 0\).
5. For \(\theta = 3\pi/2\), \(r = 1 + \cos(3\pi/2) = 1\).
6. For \(\theta = 2\pi\), \(r = 1 + \cos(2\pi) = 2\).
By plotting these (and other intermediate) points, we get a more accurate sketch of the curve. This particular polar curve resembles a heart-shaped figure known as a cardioid.
**Setting Up the Integral:**
To determine the area enclosed by this polar curve, use the formula for the area \( A \) enclosed by a polar curve:
\[ A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta \]
For our curve \( r = 1 + \cos(\theta) \), the integral bounds will be from \( 0 \) to \( 2\pi \):
\[ A = \frac{1}{2} \int_{0}^{2\pi} (1 + \cos(\theta))^2 \, d\theta \]
Therefore, the integral setup to find the enclosed area is:
\[ A = \frac{1}{2} \int_{0}^{2\pi} (1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60ebfd9f-229f-4b12-9b1b-0e6bba34204f%2Fe3d2679f-d13d-48a8-8983-12048bc6cc84%2Fir00qru_processed.png&w=3840&q=75)
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