Respuesta :
The area of the region cut from the first quadrant by the cardioid is 3π/8 +1
A cardioid is a two-dimensional plane figure that has a heart-shaped curve. The equation of cardioid is r = a(1 ± sinθ) .
The area of the polar function can be calculated if the boundary condition that form the reason is given or can be found from the given information now we use the formula using the polar formula of the area as:
[tex]A = \int\limits^b_a {\frac{r^{2}}{2}} d\theta[/tex]
The Area of the region cut from the first quadrant by cardioid,
[tex]A = \int\limits^\frac{\pi}{2}_0 {\frac{r^{2}}{2}} d\theta[/tex]
Substituting r = 1 + sin ∅
[tex]A = \int\limits^\frac{\pi}{2}_0 {\frac{(1+ sin\theta)^{2}}{2}} d\theta[/tex]
Opening square ,
[tex]A = \frac{1}{2}\int\limits^\frac{\pi}{2}_0 {1 +2sin\theta + sin^{2}\theta d\theta[/tex]
=> [tex]A = [ \frac{1}{2} (\theta - 2cos\theta + \frac{1}{2} (\theta - \frac{1}{2}sin2\theta))]^\frac{1}{2}_{0}[/tex]
=> 3π / 8 - (-1)
=> 3π/8 +1
To know more about Cardioid here
https://brainly.com/question/12075566
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