Question

2) Find the area bounded by the curve x = y² + 2y and the line x = 3.

Answer 1

$$y^2+2y=3$$

$$y^2+2y+1=4$$

$$(y+1)^2=4$$

$$y_1=-3, y_2=1$$

$$A=\displaystyle\int_{-3}^{1}(3-(y^2+2y))dy$$

$$=\big[3y-\dfrac{y^3}{3}-y^2\big]\begin{matrix} 1 \\ -3 \end{matrix}$$

$$=3-\dfrac{1}{3}-1+9-9+9=\dfrac{32}{3}({units}^2)$$