Question

The area of a circle is a function of its radius, A=πr^2
(a) Sketch two circles, one with radius r and the second with radius r + Δr, where Δr is
reasonably small.
(b) Express ΔA and dA symbollically.
(c) Shade the area representing ΔA on your sketch.
(d) Explain why dA is a good approximation to ΔA

Answer 1

\[\Delta A=A(r+\Delta r)-A(r) =\pi (r+\Delta r)^2 -\pi (r)^2=\\pi (2r \Delta r +\Delta r^2 )\\if\ \Delta r \rightarrow 0 \\pi (2r \Delta r +\Delta r^2 ) \approx (2r \Delta r )\\ \Delta r=dr \\ \frac{ dA }{ dr }=2\pi r\\so\\dA=2\pi r dr\]

Answer 2

when x-->0
x^2<<x
so
x^2+x =x(x+1) approximately =x(0+1)=x

Answer 3

thanks is this all a,b,c,and d?

Answer 4

yes
but can you find out ?