Question

Find the area enclosed by the parabolas y = 5x² and y = 2x² + 9.

Answer 1

$$\begin{gathered} y=5x^2\mathrm{represents}\mathrm{a}\mathrm{parabola}\mathrm{with}\mathrm{vertex}\mathrm{at}O\left(0,0\right)\mathrm{and}\mathrm{opening}\mathrm{upwards},\mathrm{symmetrical}\mathrm{about}+\mathrm{ve}\mathrm{y}\mathrm{axis} \\ \mathrm{y}=2x^2+9\mathrm{represents}\mathrm{the}\mathrm{wider}\mathrm{parabola},\mathrm{with}\mathrm{vertex}\mathrm{at}\mathrm{C}\left(0,9\right) \\ \mathrm{To}\mathrm{find}\mathrm{point}\mathrm{of}\mathrm{intersection},\mathrm{solve}\mathrm{the}\mathrm{two}\mathrm{equations} \\ 5x^2=2x^2+9 \\ \Rightarrow 3x^2=9 \\ \Rightarrow x=\pm \sqrt{3} \\ \Rightarrow y=15 \\ \mathrm{Thus}A\left(\sqrt{3},15\right)\mathrm{and}A\text{'}\left(-\sqrt{3},15\right)\mathrm{are}\mathrm{points}\mathrm{of}\mathrm{intersection}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{parabolas}. \\ \mathrm{Shaded}\mathrm{area}A\text{'}OA=2\times \mathrm{area}\left(OCAO\right) \\ \mathrm{Consider}\mathrm{a}\mathrm{vertical}\mathrm{stip}\mathrm{of}\mathrm{length}=\left|y_2-y_1\right|\mathrm{and}\mathrm{width}=dx \\ \mathrm{Area}\mathrm{of}\mathrm{approximating}\mathrm{rectangle}=\left|y_2-y_1\right|dx \\ \mathrm{The}\mathrm{approximating}\mathrm{rectangle}\mathrm{moves}\mathrm{from}\mathit{}x=0\mathrm{to}\mathit{}x=\sqrt{3} \\ \therefore \mathrm{Area}\left(\mathrm{OCAO}\right)={\int }_0^{\sqrt{3}}\left|y_2-y_1\right|dx={\int }_0^{\sqrt{3}}\left(y_2-y_1\right)dx\left\{\because \left|y_2-y_1\right|=y_2-y_1\mathrm{as}{y}_2>y_1\right\} \\ ={\int }_0^{\sqrt{3}}\left(2x^2+9-5x^2\right)dx \\ ={\int }_0^{\sqrt{3}}\left(9-3x^2\right)dx \\ ={\left[\left(9x-3\frac{x^3}{3}\right)\right]}_0^{\sqrt{3}} \\ =9\sqrt{3}-3\sqrt{3} \\ =6\sqrt{3}\mathrm{sq}\mathrm{units} \\ \therefore \mathrm{Shaded}\mathrm{area}\mathrm{B}\text{'}\mathrm{A}\text{'}\mathrm{AB}=2\mathrm{area}\mathrm{OCAO}=2\times 6\sqrt{3}=12\sqrt{3}\mathrm{sq}\mathrm{units} \\ \mathrm{Thus}\mathrm{area}\mathrm{enclosed}\mathrm{by}\mathrm{two}\mathrm{parabolas}=12\sqrt{3}\mathrm{sq}\mathrm{units} \end{gathered}$$