Question

using python 3 notebook

only 4.2

An ellipse is defined by the equation which is solved for y to give a function yx) for the ellipse. 4.1 " Define a function that takes x, a, b as input and returns y Make a plot of the full ellipse including all four quadrants using a-2 and b-1. Use a blue, dashed line. 4.2 Calculate the area under a quarter of the ellipse, use the top-right quadrant by approximating the quarter ellipse with vertical rectangles. The side length of each rectangle is Ax, while Ay follows from the formula for the circumference. Make sure that you calculate Δ y for each rectangle for the X value centered in the x direction in each Δ. 1. Make a bar plot of the rectangles and overplot the line of the ellipse to show this situation exactly. Hint: part of each rectangle should be outside the ellipse, and small areas inside the ellipse are not covered 2. Now calculate the area. Do not use a for loop but instead use only vector operations. Verify that you have the right answer by comparing it with the value computed from the formula for the area of the lipse Aellipexab 3. Calculate the difference between the sum of the rectangles and the value from the formula for 10 and 30 rectangles and save these differences in the variables dA10 and dA38. 4. Open a file called ellipse.dat and write two rows with two columns. The columns are the number of rectangles and the differences dA calculated in 3. Use formatted output: 2 digits for the integer, one space. 7 digits for the float with 5 after the decimal point. Close the file check the output and add it to your assignment commit. 5. Comment on the values dA18 and da3e. Why are they different?

Answer 1

as per HOMEWORKLIB RULES, i have answered only 4.1, post again for the rest

Below is the full code for your question.

I will provide the complete description at the bottom.

import numpy as np
import matplotlib.pyplot as plt

def func(x,a,b):
return (1-((x**2)*(b**2))/a**2)**0.5

X = np.linspace(-2,2,20)
X = X.tolist()
y = [func1(x,2,1) for x in X]
y += [-func1(x,2,1) for x in lin[::-1]]

X += X[::-1]

plt.plot(X,y)

output :

I have defined the function for your provided equation. See that in the function ** is an exponentiation operator. so for square a value we make a**2 and for square root of a variable  we make a**0.5.

Now to plot the graph, I have taken 20 linearly separated values between -2 and 2 as x value. I have taken -2 and 2 because given a is 2. so the x value will range from -2 to 2. Next I have calculated the values of y for each values of x. As the output of function gives only positive numbers, I have also taken the negative values for y so that we will get the full ellipse. a[::-1] will give the reversed list. Next I have used plot function from plt to plot the values of X and y.