3137 These exercises assume familiarity with the basic conceptsof parametric curves. If

Chapter 6, Problem 31

(choose chapter or problem)

These exercises assume familiarity with the basic concepts of parametric curves. If needed, an introduction to this material is provided in Web Appendix I.

For these exercises, divide the interval \([a, b]\) into \(n\) subintervals by inserting points \(t_{1}, t_{2}, \ldots, t_{n-1}\) between \(a=t_{0}\) and \(b=t_{n}\), and assume that \(x^t(t)\) and \(y^t(t)\) are continuous functions and that no segment of the curve

\(x=x(t),\quad\ \ \ \ \ \ y=y(t)\quad\ \ \ \ \ \ (a\le t\le b)\)

is traced more than once.

Let \(S\) be the area of the surface generated by revolving the curve \(x=x(t),\ y=y(t)\ (a\le t\le b)\) about the x-axis. Explain how \(S\) can be approximated by

\(S \approx \sum_{k=1}^{n}\left(\pi\left[y\left(t_{k-1}\right)+y\left(t_{k}\right)\right] \times \sqrt{\left[x\left(t_{k}\right)-x\left(t_{k-1}\right)\right]^{2}+\left[y\left(t_{k}\right)-y\left(t_{k-1}\right)\right]^{2}}\right.\)

Using results from advanced calculus, it can be shown that as max \(\Delta \mathrm{t}_{\mathrm{k}} \rightarrow 0\), this sum converges to

\(S=\int_{a}^{b} 2 \pi y(t) \sqrt{\left[x^{\prime}(t)\right]^{2}+\left[y^{\prime}(t)\right]^{2}} d t\)

Equation Transcription:

  

Text Transcription:

[a, b]

n

t_1,t_2,…,t_n-1

a=t_0

b=t_n

x^t (t)

y^t (t)

x=x(t),   y=y(t)    (a leq t leq b)

S

x=x(t),  y=y(t) (a leq t leq b)

S

S approx sum_k=1^n (pi[y(t_k-1)+y(t_k)] x sqrt [x(t_k)-x(t_k-1)]^2 + [y(t_k)-y(t_k-1)]^2

Delta t_k rightarrow 0

S=int_a^b 2 pi y(t) sqrt [x^prime(t)]^2 + [y^prime(t)]^2 dt