Let $f: [0,1] \to \mathbb{R}$ a continuous, differentiable function with $f \ge 0$. Rotate the graph of $f$ around the x-axis. Define this rotation body in $\mathbb{R}^3$ with $A$ and the area in $\mathbb{R}^2$ between the x-axis and the graph of $f$ with $B$.
(a) Now enlarge $B$ around $(0,0)$ with factor $r >0$, so the new set is $rB:= \{r\cdot(x,y) : (x,y)\in B\}$. Ascertain the area and perimeter of $rB$!
(b) Enlarge $A$ around $(0,0,0)$ with factor $r>0$, so the new set is $rA:= \{r\cdot(x,y,z) : (x,y,z)\in A\}$. Ascertain volume and surface area of $rB$.
I haven't any idea.. unfortunately the concent doesn't fit to the lecture.. Any help and ideas?
I think what the problem is trying to say is that if the point $(x,f(x))\in B$, then $(rx,rf(x))\in rB$, so we can parameterize $rB$ as $$\left(u,rf\left(\frac ur\right)\right)$$ For $u\in[0,r]$. Then the perimeter of the region bounded by $u=0$, $y=0$, $u=r$, and $y=rf\left(\frac ur\right)$ is $$\begin{align}P(r)&=rf(0)+r+rf\left(\frac rr\right)+\int_0^r\sqrt{1+\left(\frac d{du}\left(rf\left(\frac ur\right)\right)\right)^2}du\\ &=rf(0)+r+rf(1)+\int_0^r\sqrt{1+\left(f^{\prime}\left(\frac ur\right)\right)^2}du\\ &=rf(0)+r+rf(1)+\int_0^1\sqrt{1+\left(f^{\prime}\left(x\right)\right)^2}r\,dx=rP(1)\end{align}$$ Where we have made the substitution $u=rx$. Also $$A(r)=\int_0^rrf\left(\frac ur\right)du=\int_0^1rf(x)\cdot r\,dx=r^2A(1)$$ For the surface area of the solid of revolution we have $$\begin{align}S(r)&=\pi(rf(0))^2+\pi\left(rf\left(\frac rr\right)\right)^2+\int_0^rrf\left(\frac ur\right)\sqrt{1+\left(\frac d{du}\left(rf\left(\frac ur\right)\right)\right)^2}du\\ &=\pi(rf(0))^2+\pi(rf(1))^2+\int_0^rrf\left(\frac ur\right)\sqrt{1+\left(f^{\prime}\left(\frac ur\right)\right)^2}du\\ &=\pi(rf(0))^2+\pi(rf(1))^2+\int_0^1rf\left(x\right)\sqrt{1+\left(f^{\prime}\left(x\right)\right)^2}r\,dx=r^2S(1)\end{align}$$ And its volume, $$V(r)=\int_0^r\pi\left(rf\left(\frac ur\right)\right)^2du=\int_0^1\pi\left(rf(x)\right)^2\cdot r\,dx=r^3V(1)$$ So they are just trying to get you to show that blowing everything up by a factor of $r$ blows up the perimeter by $r$, areas by $r^2$, and the volume by $r^3$.