Exercise 3: (4 mark) - Consider the integral: \( \boldsymbol{I}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \boldsymbol{e}^{-\mathbf{1 0}\left(\boldsymbol{x}^{2}+\boldsymbol{y}^{2}\right)} \boldsymbol{d x} \boldsymbol{d} \boldsymbol{y} \) 1) Write the integral I in polar coordinates \( (r, \theta) \) 2) Calculate the value of the integral I from its polar expression.

To convert the integral \( I = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-10(x^2 + y^2)} dx dy \) to polar coordinates, we need to express the integral in terms of \( r \) and \( \theta \). 1. **Converting the integral to polar coordinates:** In polar coordinates, \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). The integral becomes: \[ I = \int_{0}^{2\pi} \int_{0}^{\infty} e^{-10r^2} r dr d\theta \] 2. **Calculating the value of the integral:** To calculate the value of the integral, we can use the following steps: - First, integrate with respect to \( r \) from \( 0 \) to \( \infty \). - Then, integrate with respect to \( \theta \) from \( 0 \) to \( 2\pi \). Let's calculate the integral step by step. Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{2\pi } \int_{0}^{+\infty} e^{-10r^{2}}r dr d\theta\) - step1: Multiply the terms: \(\int_{0}^{2\pi } \int_{0}^{+\infty} re^{-10r^{2}} dr d\theta\) - step2: Evaluate the integral: \(\int_{0}^{2\pi } \frac{1}{20} d\theta\) - step3: Evaluate the integral: \(\int \frac{1}{20} d\theta\) - step4: Evaluate the integral: \(\frac{1}{20}\theta \) - step5: Return the limits: \(\left(\frac{1}{20}\theta \right)\bigg |_{0}^{2\pi }\) - step6: Calculate the value: \(\frac{\pi }{10}\) The value of the integral \( I = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-10(x^2 + y^2)} dx dy \) in polar coordinates is \( \frac{\pi}{10} \).