সৃজনশীলঃ ‘দৃশ্যকল্প: f(x)=x216−x2;g(x)=y2−8xh(x)−=x2−8y…’ যশোর বোর্ড ২০২১ – যোগজীকরণ, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

দেওয়া আছে, $f\left(x\right)=x^2\sqrt{16-x^2}$

$\begin{array}{cc}& {\int }_{-2}^{2}f\left(x\right)dx={\int }_{-2}^2{x}^2\sqrt{16-x^2}dx\\ =& {\int }_{-\frac{\pi }{6}}16\sin \theta \cdot 4\sqrt{1-{\sin }^2\theta }4\cdot \cos \theta d\theta \\ & \frac{\pi }{6}\\ =& {\int }_{\frac{\pi }{6}}256{\sin }^2\theta \cdot {\cos }^2\theta d\theta \end{array}$

$\begin{array}{cc}& =64{\int }_{\frac{\pi }{6}}^{\frac{\pi }{6}}(\sin 2\theta )d\theta \\ =& \frac{64}{2}{\int }_{\frac{-\pi }{6}}^{\frac{\pi }{6}}2{\sin }^2\theta d\theta \\ =& 32{\int }_{\frac{\pi }{6}}(1-\cos 4\theta )d\theta \end{array}$

$=32{[\theta -\frac{\sin 4\theta }{4}]}_{\frac{-\pi }{6}}^{\frac{\pi }{6}}$

$=32[\frac{\pi }{6}-\frac{1}{4}\cdot {\sin }^2\frac{2\pi }{3}+\frac{\pi }{6}+\frac{1}{4}\sin \left(\frac{-2\pi }{3}\right)]$

$=32[\frac{\pi }{3}-\frac{1}{4}\cdot \frac{\sqrt{3}}{2}-\frac{1}{4}\cdot \frac{\sqrt{3}}{2}]$

$=32[\frac{\pi }{3}-\frac{2\sqrt{3}}{8}]$

$=\frac{32\pi }{3}-8\sqrt{3}$

$g\left(x\right)=y^2-8x$

$h(x{)}_{-}=x^2-8y$

$g\left(x\right)=0$ এবং $h\left(x\right)=0$

$\therefore y^2-8x=0$

$\Rightarrow y^2=8x$

$\Rightarrow y=2\sqrt{2}\sqrt{x}$

এবং $x^2=8y$

$\Rightarrow y=\frac{x^2}{8}$

মনে করি,

$y_1=2\sqrt{2}\sqrt{x}$

$y_2=\frac{x^2}{8}$

এখন,

$2\sqrt{2x}=\frac{x^2}{8}$

$\Rightarrow 8x=\frac{x^4}{64}$

$\Rightarrow 512x=x^4$

$\Rightarrow x(x^3-512)=0$

$\Rightarrow x=0,8$

$\therefore x=0$ হলে $y=0$

এবং $x=8$ হলে $y=8$

পরাবৃত্তদ্বয়ের ছেদবিন্দু $O(0,0)$ এবং $A(8,8)$

$\therefore$ নির্ণেয় ক্ষেত্রফল,

${\int }_0^8(y_1-y_2)dx$

$={\int }_0^8(2\sqrt{2}\sqrt{x}-\frac{x^2}{8})dx$

$=2\sqrt{2}{\int }_0^8\sqrt{x}dx-\frac{1}{8}\int x^{2}dx$

$=2\sqrt{2}\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]0-\frac{1}{8}\left[\frac{x^3}{3}\right]\begin{array}{cc}& 8\\ & 0\end{array}$

$=\frac{4\sqrt{2}}{3}(8^{\frac{1}{2}}-0)-\frac{1}{24}(8^3-0)$

$=\frac{4\sqrt{2}}{3}\times 16\sqrt{2}-\frac{64}{3}$

$=\frac{64}{3}$ বর্গ একক।