I can analyze the assignment you uploaded. Let me extract the content from the image and provide the detailed solution to the questions. One moment, please! PIL import Image import pytesseract

Load the image

image_path = "/mnt/data/17387617691672by87fx9.jpg" image = Image.open(image_path)

Extract text using pytesseract

extracted_text = pytesseract.image_to_string(image) extracted_textHere is the extracted content from your uploaded assignment. The questions are as follows:

Question 1: (a) Change the order of integration
$\int_0^1 \int_x^{\sqrt{2 - x^2}} \frac{x}{\sqrt{x^2 + y^2}} \, dy \, dx$
and hence evaluate. (8 Marks)

(b) Find the area lying inside the cardioid $r = a(1 + \cos\theta)$ and outside the circle $r = a$. (7 Marks)

Question 2: (a) Change the integral
$\int_0^2 \int_0^{\sqrt{4 - x^2}} (x^2 y + y^3) \, dy \, dx$
into polar coordinates and hence evaluate. (8 Marks)

(b) Evaluate
$\int_0^{\ln(8)} \int_0^2 \int_x^{x + \ln(8)} e^{x + y + z} \, dz \, dy \, dx$
(7 Marks)

Would you like to start solving these questions one by one, or focus on a specific part? Let me know!

Here are 5 relevant questions to expand understanding:

1. How do you approach the transformation of integration order in double integrals?
2. What is the significance of polar coordinates in evaluating double integrals?
3. How do you find areas enclosed by curves in polar coordinates?
4. What is the role of symmetry in simplifying integrals involving circular regions?
5. How do you interpret and simplify triple integrals with exponential functions?

Tip: Visualizing the region of integration using graphs helps significantly when changing the order of integration or converting to polar coordinates.