Vertical symmetry because you can cut a square in half vertically and the two parts be exactly the same. Horizontal symmetry because you can cut a square in half horizontally and the two parts be exactly the same. Diagonal symmetry because you can cut the square from corner to corner and the two parts look exactly the same. Rotational symmetry because you can rotate the square and it still look exactly the same. So a square has all of those symmetries. I hope this helped!
The trapezoidal approximation will be the average of the left- and right-endpoint approximations.
Let's consider a simple example of estimating the value of a general definite integral,
![\displaystyle\int_a^bf(x)\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_a%5Ebf%28x%29%5C%2C%5Cmathrm%20dx)
Split up the interval
![[a,b]](https://tex.z-dn.net/?f=%5Ba%2Cb%5D)
into
![n](https://tex.z-dn.net/?f=n)
equal subintervals,
![[x_0,x_1]\cup[x_1,x_2]\cup\cdots\cup[x_{n-2},x_{n-1}]\cup[x_{n-1},x_n]](https://tex.z-dn.net/?f=%5Bx_0%2Cx_1%5D%5Ccup%5Bx_1%2Cx_2%5D%5Ccup%5Ccdots%5Ccup%5Bx_%7Bn-2%7D%2Cx_%7Bn-1%7D%5D%5Ccup%5Bx_%7Bn-1%7D%2Cx_n%5D)
where
![a=x_0](https://tex.z-dn.net/?f=a%3Dx_0)
and
![b=x_n](https://tex.z-dn.net/?f=b%3Dx_n)
. Each subinterval has measure (width)
![\dfrac{a-b}n](https://tex.z-dn.net/?f=%5Cdfrac%7Ba-b%7Dn)
.
Now denote the left- and right-endpoint approximations by
![L](https://tex.z-dn.net/?f=L)
and
![R](https://tex.z-dn.net/?f=R)
, respectively. The left-endpoint approximation consists of rectangles whose heights are determined by the left-endpoints of each subinterval. These are
![\{x_0,x_1,\cdots,x_{n-1}\}](https://tex.z-dn.net/?f=%5C%7Bx_0%2Cx_1%2C%5Ccdots%2Cx_%7Bn-1%7D%5C%7D)
. Meanwhile, the right-endpoint approximation involves rectangles with heights determined by the right endpoints,
![\{x_1,x_2,\cdots,x_n\}](https://tex.z-dn.net/?f=%5C%7Bx_1%2Cx_2%2C%5Ccdots%2Cx_n%5C%7D)
.
So, you have
![L=\dfrac{b-a}n\left(f(x_0)+f(x_1)+\cdots+f(x_{n-2})+f(x_{n-1})\right)](https://tex.z-dn.net/?f=L%3D%5Cdfrac%7Bb-a%7Dn%5Cleft%28f%28x_0%29%2Bf%28x_1%29%2B%5Ccdots%2Bf%28x_%7Bn-2%7D%29%2Bf%28x_%7Bn-1%7D%29%5Cright%29)
![R=\dfrac{b-a}n\left(f(x_1)+f(x_2)+\cdots+f(x_{n-1})+f(x_n)\right)](https://tex.z-dn.net/?f=R%3D%5Cdfrac%7Bb-a%7Dn%5Cleft%28f%28x_1%29%2Bf%28x_2%29%2B%5Ccdots%2Bf%28x_%7Bn-1%7D%29%2Bf%28x_n%29%5Cright%29)
Now let
![T](https://tex.z-dn.net/?f=T)
denote the trapezoidal approximation. The area of each trapezoidal subdivision is given by the product of each subinterval's width and the average of the heights given by the endpoints of each subinterval. That is,
![T=\dfrac{b-a}n\left(\dfrac{f(x_0)+f(x_1)}2+\dfrac{f(x_1)+f(x_2)}2+\cdots+\dfrac{f(x_{n-2})+f(x_{n-1})}2+\dfrac{f(x_{n-1})+f(x_n)}2\right)](https://tex.z-dn.net/?f=T%3D%5Cdfrac%7Bb-a%7Dn%5Cleft%28%5Cdfrac%7Bf%28x_0%29%2Bf%28x_1%29%7D2%2B%5Cdfrac%7Bf%28x_1%29%2Bf%28x_2%29%7D2%2B%5Ccdots%2B%5Cdfrac%7Bf%28x_%7Bn-2%7D%29%2Bf%28x_%7Bn-1%7D%29%7D2%2B%5Cdfrac%7Bf%28x_%7Bn-1%7D%29%2Bf%28x_n%29%7D2%5Cright%29)
Factoring out
![\dfrac12](https://tex.z-dn.net/?f=%5Cdfrac12)
and regrouping the terms, you have
![T=\dfrac{b-a}{2n}\left((f(x_0)+f(x_1)+\cdots+f(x_{n-2})+f(x_{n-1}))+(f(x_1)+f(x_2)+\cdots+f(x_{n-1})+f(x_n))\right)](https://tex.z-dn.net/?f=T%3D%5Cdfrac%7Bb-a%7D%7B2n%7D%5Cleft%28%28f%28x_0%29%2Bf%28x_1%29%2B%5Ccdots%2Bf%28x_%7Bn-2%7D%29%2Bf%28x_%7Bn-1%7D%29%29%2B%28f%28x_1%29%2Bf%28x_2%29%2B%5Ccdots%2Bf%28x_%7Bn-1%7D%29%2Bf%28x_n%29%29%5Cright%29)
which is equivalent to
![T=\dfrac12\left(L+R)](https://tex.z-dn.net/?f=T%3D%5Cdfrac12%5Cleft%28L%2BR%29)
and is the average of
![L](https://tex.z-dn.net/?f=L)
and
![R](https://tex.z-dn.net/?f=R)
.
So the trapezoidal approximation for your problem should be
The given equation is
![x^2 + y = 144](https://tex.z-dn.net/?f=%20x%5E2%20%2B%20y%20%3D%20144%20)
First we solve for y, and for that, we have to islolate y, that is
![y= 144-x^2](https://tex.z-dn.net/?f=%20y%3D%20144-x%5E2%20)
And it pass the vertical line test. So here y is a function of x .
So we can define the given equation as y as a function of x .
So you'd have to convert all of the values to one form, and I find it easiest to do it in decimals. 61% would be 0.61, 0.605 stays the same, 3/5 would be 0.6, and 59% would be 0.59. Now you can order them:
0.59, 0.6, 0.605, 0.61
You have to convert them back to their original form, however, so your answer would be
59%, 3/5, 0.605, 61%
I hope this helps!
Step-by-step explanation:
The graph is shown in the image above.