An ellipse has equation $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ where \(b

Question

An ellipse has equation $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ where $b > a > 0$. Show that the area $A$ of the region enclosed by the ellipse is given by $A = \frac{4b}{a}\displaystyle\int_{0}^{a}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\ \mathrm{d}x$.[1] By using the substitution $x = a\sin\theta$, find $A$ in terms of $a$, $b$, and $\pi$.[4] The region enclosed by the ellipse is rotated about the $x$-axis through $\pi$ radians to form an ellipsoid. Find the volume of the ellipsoid formed, in terms of $a$, $b$, and $\pi$.[3] When a continuous function $y = \mathrm{f}(x)$, $a \leq x \leq b$, is rotated completely about the $x$-axis, the resulting curved surface area is given by $S = \displaystyle\int_{a}^{b}{2\pi y \sqrt{1+{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}}\ \mathrm{d}x}$. By considering the circle with equation ${{x}^{2}}+{{y}^{2}}={{r}^{2}}$ where $-r\leq x\leq r$, show that the surface area of a sphere with radius $r$ is $4\pi {{r}^{2}}$.[4]

Accepted Answer

(a)(i) $\pi ab\text{ units}^2$ (a)(ii) $\dfrac{4}{3}\pi ab^2\text{ units}^3$ (b) $4\pi r^2\text{ units}^2$

2019 ACJC P2 Q3

2019 ACJC P2 Q3

Video Solution:

Solution:

Solution locked