Calculus 2

Find the surface area of the solid formed by revolving the curve $x = 3\sqrt{4 - y}$ about the y-axis, where $y$ ranges from 0 to $\frac{15}{4}$.

Calculate the surface area of a solid of revolution when rotating a given curve segment about the x-axis using the formula $S = \int_{a}^{b} 2\pi y \, \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$ or $S = \int_{c}^{d} 2\pi y \, \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy$, depending on whether the curve is described as $y(x)$ or $x(y)$.

Given a function that forms a squiggly line, rotate it around the x-axis to form a three-dimensional shape. Find the surface area of this shape.

Find the surface area of the curve $x^3$ rotated around the x-axis from $x = 0$ to $x = 1$.

Find the surface area of revolution of the function f(x) = x^3 around the x-axis from x = 0 to x = 1.

Find the functions where the second derivative of the function plus two times the first derivative of the function is equal to three times the function itself.

Solve the differential equation $\frac{dy}{dx} = \frac{x^2}{y^2}$ using separation of variables to find the general solution and the particular solution given the initial condition $y(1) = 2$.

Solve the differential equation $\frac{dy}{dx} = xy$ using separation of variables, given the initial condition $y(0) = 5$, and find both the general and particular solutions.

Solve the differential equation $\frac{dy}{dx} = y^2 + 1$ and find the general solution as well as the particular solution given the initial condition $y(1) = 0$.

Given differential equations, find their order and degree.

Name the order, linearity (linear or non-linear), and homogeneity (homogeneous or non-homogeneous) of the following differential equations.

Calculate the volume of a solid of revolution by using the disc and shell methods for a given region in a plane spun about an axis.

Evaluate the integral $\displaystyle \int x e^x \, dx$ using integration by parts.

Integrate a rational function using partial fractions.

Compute the surface area of a region of revolution, specifically Gabriel's horn for $\frac{1}{x}$.

Determine the convergence of the infinite series $1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots$ and verify it is equal to $e^x$.

Convert Cartesian coordinates to polar coordinates and sketch the polar curve for $r = \cos(2\theta)$.

Find the general solution of the first-order linear differential equation $\displaystyle \frac{dy}{dx} + \frac{2}{x} y = 3x - 5$.

Find the particular solution of the differential equation with initial condition $y(e) = e$.

Solve a first order linear ordinary differential equation using the integrating factor method.