Calculus 2

Find the length of the curve $y=\frac{1}{6}x^3+\frac{1}{2x}$ from $x=1$ to $x=5$.

Find the arc length of the curve $x = \frac{2}{3}(y-1)^{3/2}$ from $y = 16$ to $y = 25$.

Find a curve through the point $(2, -4)$ whose length integral from $y = 1$ to $y = 2$ is given by $\displaystyle \int_{1}^{2} \sqrt{1 + \left( \frac{4}{y^3} \right)^2} \, dy$.

Find the length of the curve from one point to another using integration and calculus techniques for calculating arc length.

Find the arc length of the parametric curve given by $x(t) = 2 + 6t^2$ and $y(t) = 5 + 4t^3$ for $t$ in the interval $[0, \sqrt{8}]$.

Find the arc length of the parametric curve given by $x(t) = 9t^2$ and $y(t) = 9t - 3t^3$ for $t$ in the interval $[0, 2]$.

Consider the arc on the curve $y=\ln(x^2-1)$ from $x=2$ to $x=8$. Compute the following: (a) Find the arc length. (b) Find the surface area when the arc is rotated about the x-axis. (c) Find the surface area when the arc is rotated about the y-axis.

Given $x = -y^2 + 4y$, find the arc length from $y = 0$ to $y = 4$ and the surface area when the arc is rotated about the x-axis and y-axis.

Given $x^3 = y^5 + 2$, find the arc length from $y = 1$ to $y = 3$ and the surface area when the arc is rotated about the x-axis and y-axis.

Given a function, find the arc length from $x = a$ to $x = b$ using the formula for arc length $\int_a^b \sqrt{1 + (f'(x))^2} \, dx$.

Using integration, find the exact length around the curve from point A to point B for a given function f(x).

Evaluate the integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{1}{x} \, dx$ and determine if it is convergent or divergent.

Integrate $\frac{1}{x^2}$ from 1 to infinity and determine if it is convergent or divergent.

Determine if the integral of $\displaystyle \int \frac{1}{(3x + 1)^2} \, dx$ is convergent or divergent.

Evaluate the improper integral $\displaystyle \int_{1}^{\infty} \frac{1}{x^2} \, dx$.

Evaluate $\lim_{{B \to \infty}} \displaystyle \int_{{1}}^{{B}} \frac{1}{x} \, dx$.

Evaluate the limit $\lim_{{B \to \infty}} \int_{{1}}^{{B}} \frac{1}{x^n} \, dx$ for any power $n$.

Evaluate the integral $\displaystyle \int_{1}^{\infty} e^{-2x} \, dx$.

Evaluate $\displaystyle \lim_{{B \to \infty}} \int_{{0}}^{{B}} \cos x \, dx$.

Evaluate $\lim_{{a \to -\infty}} \int_{{a}}^{{0}} e^x \, dx$ using integration by parts.