Arc Length and Surface Area Assignment.sagews

Arc Length and Surface Area Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Approximate the length of $f(x)=x^4$ from $x=0$ to $x=1$ using the following steps:

• Divide the interval $[0,1]$ into $5$ equal subdivisions of width $\Delta x=\frac{1}{5}$.

• Calculate the length of the line segment from $(x_i,f(x_i))$ to $(x_{i+1},f(x_{i+1}))$ using the distance formula for $i=0,\ 1,\ 2,\ 3,\ 4$.

• Add up the five lengths from the last step. This is your approximation. Convert your approximation to a decimal using N(_).

• Now use numerical_integral to calculate the actual arc length.

• Calculate the difference between your approximation and the actual value (this is your error). [Answer: error $\approx 0.0059$]

Question 2

Find the length of $\displaystyle f(x)=\sqrt{1-x^2}$ from $x=-1$ to $x=1$. [Answer: $\pi$]

Question 3

Find the area of the surface formed by rotating around the $x$-axis the graph of $\displaystyle f(x)=\sin(x)$ from $x=0$ to $x=\pi$. [Answer: $\approx 14.42$]

Question 4

Find the area of the surface formed by rotating around the $y$-axis the graph of $\displaystyle g(y)=y^3$ from $y=1$ to $y=2$. [Answer: $\approx 199.48$]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.