Derive ellipse area, ellipsoid volume and sphere surface area

(a) The ellipse $x^2/a^2+y^2/b^2=1$ has $b>a>0$.

(i) Show its enclosed area is $A=\tfrac{4b}{a}\int_0^a\sqrt{a^2-x^2}\,dx$. 1 mark Use $x=a\sin\theta$ to find $A$ in terms of $a,b,\pi$.4 marks

(ii) Rotate the enclosed region through $\pi$ radians about the x-axis to form an ellipsoid. Find its volume.3 marks

(b) The curved surface area generated by completely rotating a nonnegative curve $y=f(x)$, $a\le x\le b$, about the x-axis is

$$S=\int_a^b2\pi y\sqrt{1+(dy/dx)^2}\,dx.$$

Use the circle $x^2+y^2=r^2$, $-r\le x\le r$, to show the surface area of a sphere is $4\pi r^2$.4 marks