Answer to Question #340673 in Calculus for Jane

Question

Find the area, take the elements of the area perpendicular to the x-axis. x²-y+1=0; x-y+1=0.

Answer 1

$$x^2-y+1=0=>y=x^2+1$$

$$x-y+1=0=>y=x+1$$

$$x^2+1=x+1$$

$$x^2=x$$

$$x_1=0, x_2=1$$

$$Area=A=\displaystyle\int_{0}^{1}(x+1-(x^2+1))dx$$

$$=\displaystyle\int_{0}^{1}(x-x^2)dx=[\dfrac{x^2}{2}-\dfrac{x^3}{3}]\begin{matrix} 1 \\ 0 \end{matrix}=\dfrac{1}{6}({unis}^2)$$

Area is $\dfrac{1}{6}$ square units.