Answer:
The answer to the question = pi/2
Step-by-step explanation:
A detailed step by step workings has been attached below.
After converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
When the Cartesian coordinates (x,y) are expressed in the polar coordinates (r, θ), then this form is called the polar form.
The given integral function in the problem is,
[tex]\int_0^{\sqrt{4}} \int\limits^x_{-x} dydx[/tex]
Let suppose, [tex]x=r\cos\theta[/tex] and [tex]y=r\sin\theta[/tex]. Thus,
[tex]\sin\theta=\dfrac{y}{r}\\\cos\theta=\dfrac{x}{r}[/tex]
Limits are y=x. From the trigonometry, the value of theta in the given triangle can be given as,
[tex]\dfrac{\sin\theta}{\cos\theta}=1\\\tan\theta=1\\\theta=\tan^{-1}1\\\theta=45^o\\\theta=\dfrac{\pi}{4}[/tex]
Similarly, for y=-x the value of angle,
[tex]\theta=-\dfrac{\pi}{4}[/tex]
Thus, the limits of theta are from -π/4 to π/4. From the Pythagoras theorem,
[tex]r^2=x^2+y^2\\r^2=(r\cos\theta)^2+(r\sin\theta)^2\\r^2=r^2(1)[/tex]
Thus, the limits of r is from 0 to 1. Convert the given integral in polar form as,
[tex]\int\limits^{\pi/4}_{-\pi/4} \int_0^{1} dt ds\\\int\limits^{\pi/4}_{-\pi/4} [1-0] dt \\\int\limits^{\pi/4}_{-\pi/4} dt \\\dfrac{\pi}{4}-\left(-\dfrac{\pi}{4} \right) \\\dfrac{\pi}{2} \\[/tex]
Hence, after converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
Learn more about the polar form here;
https://brainly.com/question/24943387
