# Volumes for Two Functions

Home If the region bounded by the upper function $y_{upper}=f(x)$ and the lower funciton $y_{lower}=g(x)$, and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, then its volume of revolution is given by: \begin{align} \displaystyle V &= \int_{a}^{b}{\Big([f(x)]^2 - [g(x)]^2\Big)}dx \\ &= \int_{a}^{b}{\Big(y_{upper}^2 - y_{lower}^2\Big)}dx \end{align} Example 1 Find the volume of [...]

# Integration of Exponential Functions

Home The base formula of integrating exponential function is obtained from deriving $e^x$. \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} This base formula is extended to the following general formula.  \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} \times a [...]

# Basic Integration Rules

Home Antiderivatives In many cases in calculus, it is known that the rate of change of one variable with respect to another, but we do not have a formula which relates the variables. In other words, it is known that $\dfrac{dy}{dx}$, but we need to know $y$ in terms of $x$. The process of finding [...]

# Calculation of Areas under Curves

Home Consider the function $f(x)=x^2+2$. We wish to estimate the green area enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$. Suppose we divide the $x$-interval into three strips of width 1 unit. Upper Rectangles The diagram below shows upper rectangles, which are rectangles with top edges at the maximum value of [...]