Honors Calculus Notes for [6 Applications of Integration]

Definition

Def. 6.1 Arc Length 弧长

Consider a curve \(C\) defined by a continuous function \(y=f(x)\) on the interval \([a, b]\). Let \(P\) be a partition of the interval \([a, b]\):

\[P: a=x_0\leq x_1\leq \cdots\leq x_{n-1}\leq x_{n}=b. \]

For each \(i\) where \(0\leq i\leq n\), we denote the point \((x_i, f(x_i))\) as \(P_i\).

We approximate the curve \(C\) by a series of line segments \(\lbrace P_{i-1}P_{i}\rbrace\) and define the arc length \(L\) of the curve \(C\) as

\[L\triangleq \lim_{||P||\rightarrow 0}\sum_{i=1}^{n}|P_{i-1}P_{i}|. \]

Def. 6.2 Surface area of revolution 旋转体表面积

Consider the surface generated by rotating the curve defined by a continuous and positive function \(y=f(x)\) around the \(x\)-axis over the interval \([a, b]\). Let \(P\) be a partition of the interval \([a, b]\):

\[P: a=x_0\leq x_1\leq \cdots \leq x_{n-1}\leq x_{n}=b \]

For each \(i (0\leq i\leq n)\), we denote the point \((x_i, f(x_i))\) as \(P_{i}\). The protion of the surface between \(x_{i-1}\) and \(x_{i}\) is approximated by the band with slant height (斜高) \(|P_{i-1}P_{i}|\). Thus, we define

\[S=\lim_{||P||\rightarrow 0}\sum_{i=1}^{n}S_{i}. \]

where \(S\) is the surface area (表面积) obtianed by rotating the curve \(y=f(x)\) from \(a\) to \(b\) about the \(x\)-axis, and \(S_i\) represents the area of the band with slant height \(|P_{i-1}P_{i}|\).

Def. 6.3 Volume 体积

• In the Horizontal Direction

  Let \(S\) be a solid that extends between the planes \(x=a\) and \(x=b\). The cross-sectional (截面) area of \(S\) at the plane \(x=t\), which is perpendicular (垂直) to the \(x\)-axis, is given by \(A(t)\), where \(A\) is a continuous function. Consider a partition \(P\) of the interval \([a, b]\):

  \[P: a=x_0\leq x_1\leq \cdots\leq x_{n-1}\leq x_n=b. \]

  Let \(x_{i}^{*}\) is any point chosen within the subinterval \([x_{i-1}, x_{i}]\). The volume \(V\) of the solid \(S\) can be expressed as

  \[V=\lim_{||P||\rightarrow 0}\sum_{i=1}^{n}A(x_{i}^{*})\Delta x_i=\int_{a}^{b}A(x)\text{d}x \]

• In the Vertical Direction

  If \(S\) is a solid that lies between \(y = c\) and \(y = d\) and if the cross-sectional area of \(S\) through \(y\) perpendicular to the \(y\)-axis is \(A(y)\), where \(A\) is a continuous function, then, similarly, the volume of \(S\) is

  \[V=\int_{c}^{d}A(y)\text{d}y \]

Def. 6.4 Polar coordinates and polar curves 极坐标，极曲线

A point \((x, y)\) in Cartesian coordinates can be represented in polar coordinates with the following equations:

\[x=r\cos \theta, y=r\sin \theta \]

Conversely, we can express \(r\) and \(\theta\) in the term of \(x\) and \(y\) as follows:

\[r^2=x^2+y^2. \tan\theta=\frac{y}{x} \]

We often use the ordered pair \((r, \theta)\), for a point in polar coordinates, where \(r\) represents the distance from a reference point called the pole (极点), and \(\theta\) the point’s direction from the pole relative to the direction of the polar axis (极轴), a ray drawn from the pole.

A polar curve (极曲线) is a graph of a function defined in polar coordinates, where points are represented by a distance \(r\) from a fixed point (the pole) and an angle \(\theta\) from a fixed direction. The curve is described by a polar equation (极方程) of the form \(r=f(\theta)\), where \(f\) is a function that determines the distance \(r\) for each angle \(\theta\).

Proposition & Theorem

Pro. 6.1 Area between two curves

• In the Horizontal Direction

  The area between the curve \(y=f(x)\) and \(y=g(x)\) over \([a, b]\) is

  \[\int_{a}^{b}|f(x)-g(x)|\text{d}x \]

• In the Vertical Direction

  The area between the curve \(x=f(y)\) and \(x=g(y)\) over \([a, b]\) is

  \[\int_{a}^{b}|f(y)-g(y)|\text{d}y \]

Pro. 6.2 Area under a parametric curve

If a curve is tracced out once by the differentiable parametric equations \(x=f(t)\) and \(y=g(t)\), \(\alpha\leq t\leq \beta\), then the area under the curve is

\[\int_{\alpha}^{\beta} g(t)f^\prime(t)\text{d}t \]

Proof Pro. 6.2

Integration by substitution.

Example Pro. 6.2

\[x=r(t-\sin t), y=r(1-\cos t), t\in [0, 2\pi] \]

\[\begin{aligned} A&=\int_{0}^{2\pi}r(1-\cos t)\cdot [r(t-\sin t)]^\prime\text{d}t\\\\ &=r^2\int_{0}^{2\pi}(1-2\cos t+\cos^2t)\text{d}t\\\\ &=r^2\int_{0}^{2\pi}[1-2\cos t+\frac{1}{2}(1+\cos 2t)]\text{d}t\\\\ &=r^2[\frac{3}{2}t-2\sin t+\frac{1}{4}\sin 2t]|_{t=0}^{2\pi}=3\pi r^2 \end{aligned} \]

Pro. 6.3 Arc length of a smooth curve

• In the Horizontal Direction

  If the derivative \(f^\prime\) is continuous on the interval \([a, b]\), then the length of the curve defined by \(y=f(x)\) for \(a\leq x\leq b\) is given by

  \[L=\int_{a}^{b}\sqrt{1+[f^\prime(x)]^2}\text{d}x. \]

• In the Vertical Direction

  If the derivative \(f^\prime\) is continuous on the interval \([a, b]\), then the length of the curve defined by \(x=f(y)\) for \(a\leq y\leq b\) is given by

  \[L=\int_{a}^{b}\sqrt{1+[f^\prime(y)]^2}\text{d}y. \]

• Define the arc length function:

  \[s(x)=\int_{a}^{x}\sqrt{1+[f^\prime(t)]^2}\text{d}t \]

  which represents the distance along the curve from the initial point \((a, f(a))\) to the point \((x, f(x))\).

Proof Pro. 6.3

Let \(\Delta y_i=y_i-y_{i-1}=f(x_i)-f(x_{i-1})\). Then

\[|P_{i-1}P_{i}|=\sqrt{(x_i-x_{i-1})^2+(y_i-y_{i-1})^2}=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2} \]

By the Mean Value Theorem, we have

\[\Delta y_{i}=f(x_i)-f(x_{i-1})=f^\prime(x^{*}_i)(x_i-x_{i-1})=f^\prime(x^{*}_i)\Delta x_i \]

for some \(x_i^{*}\in[x_{i-1}, x_i]\).

Thus

\[\begin{aligned} |P_{i-1}P_{i}|&=\sqrt{(\Delta x_i)^2+(\Delta y_i)^2}\\\\ &=\sqrt{(\Delta x_i)^2+[f^\prime(x^{*}_{i})\Delta x_i]^2}\\\\ &=\sqrt{1+[f^\prime(x^{*}_{i})]^2}\Delta x_i \end{aligned} \]

Hence

\[\begin{aligned} L&=\lim_{||P||\rightarrow 0}\sum_{i=1}^{n}|P_{i-1}P_{i}|\\\\ &=\lim_{||P||\rightarrow 0}\sum_{i=1}^{n}\sqrt{1+[f^\prime(x^{*}_{i})]^2}\Delta x_i\\\\ &=\int_{a}^{b}\sqrt{1+[f^\prime(x)]^2}\Delta x_i\\\\ \end{aligned} \]

Pro. 6.4 Arc length of a smooth parametric curve

Consider a parametric curve defined by the parametric equations

\[x=f(t), y=g(t), \alpha\leq t\leq \beta, \]

where \(f^\prime\) and \(g^\prime\) are continuous, and the curve is traced exactly once as \(t\) increases from \(\alpha\) to \(\beta\). The arc length of curve can be expressed as

\[L=\int_{\alpha}^{\beta}\sqrt{[f^\prime(t)]^2+[g^\prime(t)]^2}\text{d}t \]

Proof Pro. 6.4

Let \(P\) be a partition of \([\alpha, \beta]\):

\[P:\alpha=t_0<t_1<\cdots<t_n=\beta \]

Denote \(L_{P}\) to be the approximation of the arc length by the partition \(P\):

\[L_{P}=\sum_{i=1}^{n}|P_{i-1}P_{i}|=\sum_{i=1}^{n}\sqrt{(\Delta x_i)^2+(\Delta y_i)^2} \]

Since \(f\) and \(g\) are differentiable, by the Mean Value Theorem, we have

\[\Delta x_i=f^\prime(t_{i}^{*})\Delta t_{i}, \Delta y_i=g^\prime(t_{i}^{**})\Delta t_{i} \]

for some \(t^{*}_i, t^{**}_i\in [t_{i-1}, t_{i}]\). Thus, we can express \(L_{P}\) as

\[L_{P}=\sum_{i=1}^{n}\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2}\Delta t_i \]

which is nearly identical to the Riemann sum

\[S(P, \sqrt{(f^\prime)^2+(g^\prime)^2})=\sum_{i=1}^{n}\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2} \]

The only distinction is that the sample points for \(f^\prime\) and \(g^\prime\) may differ in \(L_{P}\). Here in \(S\), we have chosen the same sample points for \(f^\prime\).

Thus

\[\begin{aligned} \left|L_{P}-S(P, \sqrt{(f^\prime)^2+(g^\prime)^2})\right|&=\left|\sum_{i=1}^{n}\left(\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2}-\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2}\right)\Delta t_{i}\right|\\\\ &\leq \left|\sum_{i=1}^{n}\left(\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2}-\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2}\right)\right|\Delta t_{i}\\\\ &=\sum_{i=1}^{n}\frac{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2-([f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2)}{\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2}+\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2}}\Delta t_{i}\\\\ &=\sum_{i=1}^{n}\frac{|g^\prime(t_{i}^{**})-g^\prime(t_{i}^{*})|\cdot |g^\prime(t_{i}^{**})+g^\prime(t_{i}^{*})|}{\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{**})]^2}+\sqrt{[f^\prime(t_{i}^{*})]^2+[g^\prime(t_{i}^{*})]^2}}\Delta t_{i}\\\\ &\leq \sum_{i=1}^n|g^\prime(t_{i}^{**})-g^\prime(t_{i}^{*})|\Delta t_i\\\\ &\leq \sum_{i=1}^n\omega_{[t_{i-1}, t_{i}]}(g^\prime)\Delta t_i \end{aligned} \]

Since \(g^\prime\) is integrable, \(\lim_{||P||\rightarrow 0}\sum_{i=1}^n\omega_{[t_{i-1}, t_{i}]}(g^\prime)\Delta t_i=0\).

Thus

\[\lim_{||P||\rightarrow 0}L_{P}=\lim_{||P||\rightarrow 0}S(P, \sqrt{(f^\prime)^2+(g^\prime)^2})=\int_{\alpha}^{\beta}\sqrt{[f^\prime(t)]^2+[g^\prime(t)]^2}\text{d}t \]

Pro. 6.5 Smooth surface area of revolution 光滑旋转体表面积

• In the Horizontal Direction

  \[S=\int_{a}^{b}2\pi f(x)\sqrt{1+[f^\prime(x)]^2}\text{d}x \]

• In the Vertical Direction

  \[S=\int_{a}^{b}2\pi f(y)\sqrt{1+[f^\prime(y)]^2}\text{d}y \]

Pro. 6.6 Surface area of revolution pf a parametric curve 光滑参数曲线的旋转体表面积

• In the Horizontal Direction

  \[S=\int_{\alpha}^{\beta}2\pi g(t)\sqrt{[f^\prime(t)]^2+[g^\prime(t)]^2}\text{d}t \]

• In the Vertical Direction

  \[S=\int_{\alpha}^{\beta}2\pi f(t)\sqrt{[f^\prime(t)]^2+[g^\prime(t)]^2}\text{d}t \]

Pro. 6.7 Volume of solid of revolution 旋转体的体积

• For a non-negative continuous function \(f\) defined on the interval \([a, b]\), if a solid is formed by rotating the region

  \[\lbrace (x, y): a\leq x\leq b, 0\leq y\leq f(x)\rbrace \]

  around the \(x\)-axis, then the volume of the solid is given by

  \[V=\int_{a}^{b}\pi[f(x)]^2\text{d}x \]

• For two non-negative continuous functions \(f\) and \(g\) satisfying \(f(x) \geq g(x)\) for all \(x \in [a,b]\), if a solid is formed by revolving the region

  \[\lbrace (x, y): a\leq x\leq b, g(x)\leq y\leq f(x)\rbrace \]

  around the \(x\)-axis, then the volume of the solid is given by

  \[V=\int_{a}^{b}\pi\lbrace[f(x)]^2-[g(x)]^2\rbrace\text{d}x \]

Pro. 6.8 Volume by cylindrical shells 圆柱壳法

The volume of the solid, obtained by rotating about the \(y\)-axis the region under the curve \(y = f(x)\) from \(a\) to \(b\) \((0 \leq a < b)\), is given by

\[V=\int_{a}^{b}2\pi xf(x)\text{d}x \]

Pro. 6.9 Arc length of polar curve 极曲线弧长

Suppose a polar curve is defined by the equation \(r=f(\theta)\). If \(f^\prime\) is continuous, then the arc length of the polar curve from \(\theta=\alpha\) to \(\theta=\beta\) is given by

\[L=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{\text{d}r}{\text{d}\theta}\right)^2}\text{d}\theta=\int_{\alpha}^{\beta}\sqrt{[f(\theta)]^2+[f^\prime(\theta)]^2}\text{d}\theta= \]

Pro. 6.10 Area of polar region 极区面积

Let \(R\) be a polar region bounded by the polar curve \(r=f(\theta)\) and by the rays from \(\theta=\alpha\) to \(\theta=\beta\), where \(f\) is a positive continuous function and the angles satisfy \(0<\beta-\alpha\leq 2\pi\). The area of the region \(R\) is given by

\[A=\int_{\alpha}^{\beta}\frac{1}{2}r^2\text{d}\theta=\int_{\alpha}^{\beta}\frac{1}{2}[f(\theta)]^2\text{d}\theta \]