Arc Length and Surface Area Assignment

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 exact value (this is your error). [Answer: 0.0059]

Question 2

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

Practice your $\LaTeX$: Write out the integral you used to compute the answer.

$\int_{-1}^1\sqrt{1+(\frac{-x}{\sqrt{1-x^2}})}$

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: 14.42]

Practice your $\LaTeX$: Write out the integral you used to compute the surface area.

$\int_0^\pi2\pi f(x)\sqrt{1+cos(x)^2}$

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: 199.48]

Practice your $\LaTeX$: Write out the integral you used to compute the surface area.

$\int_1^22\pi g(y)\sqrt{1+(3y^2)^2}$