# Finding the Area of a Figure Given by an Equation

• MHB
• WMDhamnekar
In summary, the solution provided for finding the area of the figure given by the cartesian equation involves using trigonometric substitutions to express the x and y coordinates in terms of a parameter alpha. The y-coordinate, represented by ydx, is chosen as the integrand for computing the area of the ellipse, as it represents an infinitely small rectangle with height y and width dx. By adding all these rectangles together, we get the area of the figure, which is equivalent to the integral of ydx. This is a fundamental concept in integration.
WMDhamnekar
MHB
Question:- Find the area of the figure given by the cartesian equation below:

$$\frac{(x+y)^2}{16}+\frac{(x-y)^2}{9}=1$$

Solution given:-

Let $x+y= 4\cos{\alpha},x-y=3\sin{\alpha}$ Then $x=\frac{4\cos{\alpha}+3\sin{\alpha}}{2}$ $\Rightarrow dx=\frac{3\cos{\alpha}-4\sin{\alpha}}{2}d\alpha$

$y=\frac{4\cos{\alpha}-3\sin{\alpha}}{2}$. So, $ydx=\left(3-\frac{25}{8}*\sin{2\alpha}\right)*d\alpha$ What is this ydx?

Hence the required area is

$$\displaystyle\int_0^{2\pi}\left(3-\frac{25}{8}*\sin{2\alpha}\right)*d\alpha$$ Why does the author select ydx as integrand for computation of area of ellipse? What is the logic behind that?

$=6\pi=18.849$

If any member of Math help Board knows the explanations for my queries, may reply to this question.

$y$ is the y-coordinate of a point on the edge of the figure. $dx$ is an infinitely small increment of the x-coordinate.
The product $y\,dx$ represents a rectangle of height $y$ and width $dx$. Its area is $y\,dx$.
If we add all such rectangles together, we get the area of the figure.

It's actually more or less the definition of an integral.
See here how that interpretation works.

## 1. What is the equation for finding the area of a figure?

The equation for finding the area of a figure depends on the shape of the figure. For example, the equation for finding the area of a rectangle is length x width, while the equation for finding the area of a circle is π x radius^2. It is important to know the specific equation for the shape of the figure you are working with.

## 2. How do I know which numbers to plug into the equation?

To find the area of a figure given by an equation, you will need to know the measurements of the figure. These measurements could be given to you in the problem or you may need to measure them yourself. Make sure to carefully read the problem and identify which numbers correspond to the length, width, radius, etc. that you will need to plug into the equation.

## 3. Can I use the same equation for any shape?

No, the equation for finding the area of a figure will vary depending on the shape. Each shape has its own unique formula for determining its area. It is important to know the specific formula for the shape you are working with in order to accurately find the area.

## 4. Is there a specific order in which I need to solve the equation?

For most equations, there is no specific order in which you need to solve the equation. However, it is important to follow the order of operations (PEMDAS) when evaluating the equation. This means solving any parentheses or brackets first, followed by exponents, multiplication and division from left to right, and finally addition and subtraction from left to right.

## 5. What units should the area be expressed in?

The units for area will depend on the units used for the measurements of the figure. For example, if the length and width of a rectangle are measured in inches, the area will be expressed in square inches. It is important to pay attention to the units and include them in your final answer.

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