3.8.4 Pengamiran, SPM Praktis (Kertas 2)

Soalan 7:
Rajah di bawah menunjukkan suatu lengkung $y=\frac{1}{4}{x}^2+3$ yang menyilang suatu garis lurus y = x + 6 pada titik A.

(a) Cari koordinat A.
(b) ) hitung
(i) luas rantau berlorek M,
(ii) isipadu kisaran, dalam sebutan π, apabila rantau berlorek N diputarkan melalui 360^o pada paksi-y.

Penyelesaian:
(a)
$\begin{array}{l}y=\frac{1}{4}{x}^2+\mathrm{3\dots \dots \dots .}\left(1\right)\\ y=x+\mathrm{6\dots \dots \dots .}\left(2\right)\\ \text{Gantikan (2) ke dalam (1),}\\ x+6=\frac{1}{4}{x}^2+3\\ 4x+24=x^2+12\\ x^2-4x-12=0\\ \left(x+2\right)\left(x-6\right)=0\\ x=-2\text{ or }x=6\left(\text{ditolak}\right)\\ \\ \text{Apabila }x=-2\\ y=-2+6=4\\ \text{Oleh itu, }A=\left(-2,4\right).\end{array}$


(b)(i)
$\begin{array}{l}\text{Pada paksi-}x\text{, }y=0\\ \text{Dari }y=x+6,x=-6\\ \\ \text{Luas kawasan berlorek }M\\ =\text{Luas segi tiga}+\text{Luas di bawah lengkung}\\ =\frac{1}{2}\times \left(6-2\right)\times 4+{\displaystyle {\int }_{-2}^{0}ydx}\\ =8+{\displaystyle {\int }_{-2}^0\left(\frac{1}{4}{x}^2+3\right)dx}\\ =8+{\left[\frac{x^3}{4\left(3\right)}+3x\right]}_{-2}^0\\ =8+\left[0-\left(\frac{{\left(-2\right)}^3}{12}+3\left(-2\right)\right)\right]\\ =8+\left[0-\left(-\frac{8}{12}-6\right)\right]\\ =8+\left[0-\left(-\frac{20}{3}\right)\right]\\ =14\frac{2}{3}{\text{ unit}}^2\end{array}$


(b)(ii)
$\begin{array}{l}\text{pada paksi-}y\text{, }x=0,\\ y=\frac{1}{4}\left(0\right)+3\\ y=3\\ \\ y=\frac{1}{4}{x}^2+3\\ 4y=x^2+12\\ x^2=4y-12\\ \\ \text{Isipadu }N\\ \text{= }\pi {\displaystyle {\int }_3^4{x}^{2}dy}\\ \text{= }\pi {\displaystyle {\int }_3^4\left(4y-12\right)dy}\\ \text{= }\pi {\displaystyle {\int }_3^4\left(2y^2-12y\right)dy}\\ =\pi {\left[\left(2y^2-12y\right)\right]}_3^4\\ =\pi \left[\left(2{\left(4\right)}^2-12\left(4\right)\right)-\left(2{\left(3\right)}^2-12\left(3\right)\right)\right]\\ =\pi \left(-16+18\right)\\ =2\pi {\text{ unit}}^3\end{array}$

Soalan 8:
Rajah di bawah menunjukkan lengkung $y=\frac{4}{x^2}$ dan garis lurus y = mx + c. Garis lurus y = mx + c ialah tangen kepada lengkung pada (2, 1).
(a) Cari nilai m dan nilai c.
(b) Hitung luas kawasan berlorek.
(c) Diberi bahawa isi padu kisaran apabila rantau yang dibatasi oleh lengkung, paksi-x, garis lurus x = 2 dan x = h diputarkan melalui 360^o pada paksi-x ialah $\frac{38\pi }{81}{\text{ unit}}^3.$
Cari nilai h, dengan keadaan h > 2.

Penyelesaian:
(a)
$\begin{array}{l}y=\frac{4}{x^2}=4x^{-2}\\ \frac{dy}{dx}=-8x^{-3}\\ =-\frac{8}{x^3}\\ \text{At }x=2,\\ \frac{dy}{dx}=-\frac{8}{2^3}\\ =-1\\ \\ \text{Persamaan tangen:}\\ y-y_1=m\left(x-x_1\right)\\ y-1=-1\left(x-2\right)\\ y=-x+2+1\\ y=-x+3\\ \\ m=-1,c=3\end{array}$


(b)
$\begin{array}{l}\text{Pada paksi-}x\text{, }y=0\\ \text{Dari garis lurus }y=-x+3,x=3\\ \\ \text{Luas kawasan berlorek}\\ =\text{Luas bawah lengkung}-\text{Luas segi tiga}\\ ={\displaystyle {\int }_2^{4}ydx-}\frac{1}{2}\times 1\times 1\\ ={\displaystyle {\int }_2^4\left(4x^{-2}\right)dx}-\frac{1}{2}\\ ={\left[\frac{4x^{-1}}{-1}\right]}_2^4-\frac{1}{2}\\ ={\left[-\frac{4}{x}\right]}_2^4-\frac{1}{2}\\ =\left[-\frac{4}{4}-\left(-\frac{4}{2}\right)\right]-\frac{1}{2}\\ =\frac{1}{2}{\text{ unit}}^2\end{array}$


(c)
$\begin{array}{l}\text{Isipadu kisaran}=\frac{38\pi }{81}\\ \pi {\displaystyle {\int }_2^h{y}^{2}dx}=\frac{38\pi }{81}\\ {\displaystyle {\int }_2^h{\left(\frac{4}{x^2}\right)}^{2}d}x=\frac{38}{81}\\ {\displaystyle {\int }_2^h\left(\frac{16}{x^4}\right)dx}=\frac{38}{81}\\ {\displaystyle {\int }_2^h\left(16x^{-4}\right)dx}=\frac{38}{81}\\ {\left[\frac{16x^{-3}}{-3}\right]}_2^h=\frac{38}{81}\\ {\left[-\frac{16}{3x^3}\right]}_2^h=\frac{38}{81}\\ -\frac{16}{3h^3}-\left(-\frac{16}{3{\left(2\right)}^3}\right)=\frac{38}{81}\\ \frac{16}{3h^3}=\frac{16}{24}-\frac{38}{81}\\ \frac{16}{3h^3}=\frac{16}{81}\\ 3h^3=81\\ h^3=27\\ h=3\end{array}$