Chapter 6 Applications of Integration

• Area of a Region Between Two Curves
• Volume
• Arc Length

# Area of a Region Between Two Curves

The area A of the region bounded by the curves $y =f(x), y= g(x)$ and the lines $x=a, x=b$,
where $f$ and $g$ are continuous ,is $A = \int^b_a |f(x) - g(x)|dx$

# Volume

# The Disk Method

• $V = \int^b_a A(x)dx$(about x-axis)
• $V = \int^d_c A(y)dy$(about y-axis)

# The Shell Method

• $V = \int^b_a 2 \pi xf(x) dx$ (about y-axis)
• $V = \int^d_c 2 \pi yf(y) dy$ (about x-axis)

# Arc Length

$\overline{P_iP_{i+1}} = \sqrt{(x_{i+1}-x_i)^2 + (f(x_{i+1}) - f(x_i))^2} = \sqrt{(\Delta x)^2 + (\Delta y)^2}$
$\Rightarrow \sum^n_{i=1} \overline{P_iP_{i+1}} = \sum^n_{i=1} \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sum^n_{i=1} \sqrt{(\frac{(\Delta x)^2}{(\Delta x)^2}) + \frac{(\Delta y)^2}{(\Delta x)^2}} \Delta x= \int^{x_n}_{x_i} \sqrt{1 + (f'(x))^2 } dx$

Assume that $f'$ is continuous on $[a,b]$ and $g'$ is continuous on $[c,d]$.

1. If the arc is described by $y=f(x)$,$a \le x \le b$ then $L = \int^b_a \sqrt{1+(\frac{dy}{dx})^2}dx = \int^b_a \sqrt{1+(f'(x))^2}dx$
2. If the arc is described by $x=g(y)$,$c \le y \le d$ then $L = \int^d_c \sqrt{1+(\frac{dy}{dx})^2}dy = \int^d_c \sqrt{1+(g'(y))^2}dy$

# Surface Area

Assume that $f'$ is continuous on $[a,b]$ and $g'$ is continuous on $[c,d]$.

1. If the curve is $y = f(x)$,$a \le x \le b$ and rotates about x-axis.
   $S = 2\pi f(x) \int^b_a \sqrt{1+(\frac{dy}{dx})^2} dx = 2\pi f(x) \int^b_a \sqrt{1+(f'(x))^2} dx$
2. If the curve is $x = g(y)$,$c \le x \le d$ and rotates about y-axis.
   $S = 2\pi g(y) \int^d_c \sqrt{1+(\frac{dx}{dy})^2} dy = 2\pi g(y) \int^d_c \sqrt{1+(g'(y))^2} dy$

# Reference

• 蘇承芳老師 - 微積分甲（一）109 學年度 - Calculus (I) Academic Year 109