From 7fe78254b3fea8e3a0f0b8142795334aacdaf12d Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Tue, 28 Dec 2021 16:39:26 +0000 Subject: [PATCH] add notes on integration --- .../1026_maths_for_engineering/calculus.md | 332 +++++++++++++++++- ...ot-2021-12-28T15:18:59,911868873+00:00.png | Bin 0 -> 33735 bytes ...ot-2021-12-28T15:25:12,556743251+00:00.png | Bin 0 -> 23497 bytes 3 files changed, 330 insertions(+), 2 deletions(-) create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-28T15:18:59,911868873+00:00.png create mode 100644 uni/mmme/1026_maths_for_engineering/images/vimscrot-2021-12-28T15:25:12,556743251+00:00.png diff --git a/uni/mmme/1026_maths_for_engineering/calculus.md b/uni/mmme/1026_maths_for_engineering/calculus.md index cf4694a..74bf82c 100755 --- a/uni/mmme/1026_maths_for_engineering/calculus.md +++ b/uni/mmme/1026_maths_for_engineering/calculus.md @@ -173,7 +173,7 @@ There may be more than one $y$ value for each $x$ value. -# Differentation +# Differentiation The derivative of the function $f(x)$ is denoted by: @@ -333,7 +333,7 @@ $$f'(a) = 0 \text{ and } f''(a) = 0 \text { and } f'''(a) \ne 0$$ ![](./images/vimscrot-2021-12-27T15:38:29,395666506+00:00.png) -# Taylor series +# Approximating with the Taylor series The expansion @@ -445,3 +445,331 @@ $$f(x) = f(a) + \frac 1 2 f''(a)(x-a)^2 + \cdots$$ have a maximum - If $f''(a) = 0$ then we must include a higer order terms to determine what happens have a minimum + +# Integration + +Integration is the reverse of [differentiation](#differentiation). + +Take velocity and displacement as an example: + +$$\int\! v \mathrm dt = s + c$$ + +where $c$ is the constant of integration, which is required for +[indefinite integrals](#indefinite-integrals).A + +## Definite Integrals + +The definite integral of a function $f(x)$ in the range $a \le x \le b$ is denoted be: + +$$\int^b_a \! f(x) \,\mathrm dx$$ + +If $f(x) = F'(x)$ ($f(x)$ is the derivative of $F(x)$) then + +$$\int^b_a \! f(x) \,\mathrm dx = \left[F(x)\right]^b_a = F(b) - F(a)$$ + +## Area and Integration + +Approximate the area under a smooth curve using a large number of narrow rectangles. + +![](./images/vimscrot-2021-12-28T15:18:59,911868873+00:00.png) + +Area under curve $\approx \sum_{n} f(x_n)\Delta x_n$. + +As the rectangles get more numerous and narrow, the approximation approaches the real area. + +The limiting value is denoted + +$$\approx \sum_{n} f(x_n)\Delta x_n \rightarrow \int^b_a\! f(x) \mathrm dx$$ + +This explains the notation used for integrals. + +
+ + +#### Example 1 + +Calculate the area between these two curves: + +\begin{align*} +y &= f_1(x) = 2 - x^2 \\ +y &= f_2(x) = x +\end{align*} + + + +![](./images/vimscrot-2021-12-28T15:25:12,556743251+00:00.png) + +1. Find the crossing points $P$ and $Q$ + + \begin{align*} + f_1(x) &= f_2(x) \\ + x &= 2-x^2 \\ + x &= 1 \\ + x &= -2 + \end{align*} + +2. Since $f_1(x) \ge f_2(x)$ between $P$ and $Q$ + + \begin{align*} + A &= \int^1_{-2}\! (f_1(x) - f_2(x)) \mathrm dx \\ + &= \int^1_{-2}\! (2 - x^2 - x) \mathrm dx \\ + &= \left[ 2x - \frac 13 x^3 - \frac 12 x^2 \right]^1_{-2} \\ + &= \left(2 - \frac 13 - \frac 12 \right) - \left( -4 + \frac 83 - \frac 42 \right) \\ + &= \frac 92 + \end{align*} + +
+ +## Techniques for Integration + +Integration requires multiple techniques and methods to do correctly because it is a PITA. + +These are best explained by examples so try to follow those rather than expect and explanation. + +### Integration by Substitution + +Integration but substitution lets us integrate functions of functions. + +
+ + +#### Example 1 + +Find + +$$I = \int\!(5x - 1)^3 \mathrm dx$$ + + + +1. Let $w(x) = 5x - 1$ +2. + \begin{align*} + \frac{\mathrm d}{\mathrm dx} w &= 5 \\ + \frac 15 \mathrm dw &= \mathrm dx + \end{align*} + +3. The integral is then + + \begin{align*} + I &= \int\! w^3 \frac 15 \mathrm dw \\ + &= \frac 15 \cdot \frac 14 \cdot w^4 + c \\ + &= \frac{1}{20}w^4 + c + \end{align*} + +4. Finally substitute $w$ out + + $$I = \frac{(5x-1)^4}{20} + c$$ + +
+ +
+ + +#### Example 2 + +Find + +$$I = \int\! \cos x \sqrt{\sin x + 1} \mathrm dx$$ + + + +1. Let + + $$w(x) = \sin x + 1$$ + +2. Then + + \begin{align*} + \frac{\mathrm d}{\mathrm dx} w = \cos x \\ + \mathrm dw = \cos x \mathrm dx \\ + \end{align*} + +3. The integral is now + + \begin{align*} + I &= \int\! \sqrt w \,\mathrm dw \\ + &= \int\! w^{\frac12} \,\mathrm dw \\ + &= \frac23w^{\frac32} + c + \end{align*} + +4. Finally substitute $w$ out to get: + + $$I = \frac23 (\sin x + 1)^{\frac32} + c$$ + +
+ +
+ + +#### Example 3 + +Find + +$$I = \int^{\frac\pi2}_0\! \cos x \sqrt{\sin x + 1} \,\mathrm dx$$ + + + +1. Use the previous example to get to + + $$I = \int^2_1\! \sqrt w \,\mathrm dw = \frac23w^{\frac32} + c$$ + +2. Since $w(x) = \sin x + 1$ the limits are: + + \begin{align*} + x = 0 &\rightarrow w = 1\\ + x = \frac\pi2 &\rightarrow w = 2 + \end{align*} + +3. This gives us + + $$I = \left[ \frac23w^{\frac32} \right]^2_1 = \frac23 (2^{\frac23} = 1)$$ + +
+ +
+ + +#### Example 4 + +Find + +$$I = \int^1_0\! \sqrt{1 - x^2} \,\mathrm dx$$ + + + +1. Try a trigonmetrical substitution: + + \begin{align*} + x &= \sin w \\ + \\ + \frac{\mathrm dx}{\mathrm dw} = \cos w \\ + \mathrm dx = \cos 2 \,\mathrm dw \\ + \end{align*} + +2. + + \begin{align*} + x=0 &\rightarrow w=0 \\ + x=1 &\rightarrow w=\frac\pi2 + \end{align*} + +3. Therefore + + \begin{align*} + I &= \int^{\frac\pi2}_0\! \sqrt{1 - \sin^2 w} \cos w \,\mathrm dw \\ + &= \int^{\frac\pi2}_0\! \cos^w w \,\mathrm dw + \end{align*} + + But $\cos(2w) = 2\cos^2w - 1$ so: + + $$\cos^2w = \frac12 \cos(2w) + \frac12$$ + + + Hence + + \begin{align*} + I &= \int^{\frac\pi2}_0\! \frac12 \cos(2w) + \frac12 \,\mathrm dw \\ + &= \left[ \frac14 \sin(2w) + \frac w2 \right]^{\frac\pi2}_0 \\ + &= \left( \frac14 \sin\pi + \frac\pi4 \right) - 0 \\ + &= \frac\pi4 + \end{align*} + +### Integration by Parts + +$$uv = \int\! u\frac{\mathrm dv}{\mathrm dx} \,\mathrm dx + \int\! \frac{\mathrm du}{\mathrm dx}v \,\mathrm dx$$ + +or + +$$\int\! u\frac{\mathrm dv}{\mathrm dx} \,\mathrm dx = uv - \int\! \frac{\mathrm du}{\mathrm dx}v \,\mathrm dx$$ + +This technique is derived from integrating the product rule. + +
+ +
+ + +#### Example 1 + +Find + +$$I = \int\! \ln x \,\mathrm dx$$ + + + +1. Use + + $$\int\! u\frac{\mathrm dv}{\mathrm dx} \,\mathrm dx = uv - \int\! \frac{\mathrm du}{\mathrm dx}v \,\mathrm dx$$ + +2. Set $u = \ln x$ + and $v' = 1$. + +3. This means that $u' = \frac1x$ and $v = x$. +4. + + \begin{align*} + I &= x\ln x - \int\! x\cdot\frac1x \,\mathrm dx + c \\ + &= x\ln x - \int\! \,\mathrm dx + c \\ + &= x\ln x - x + c \\ + \end{align*} + +
+ +# Application of Integration + +## Differential Equations + +Consider the equation + +$$\frac{\mathrm dy}{\mathrm dx} = y^2$$ + +To find $y$, is not a straightforward integration: + +$$y = \int\!y^2 \,\mathrm dx$$ + +The equation above does not solve for $y$ as we can't integrate the right until we know $y$... +which is what we're trying to find. + +This is an example of a first order differential equation. +The general form is: + +$$\frac{\mathrm dy}{\mathrm dx} = F(x, y)$$ + +### Separable Differential Equations + +A first order diferential equation is called *separable* if it is of the form + +$$\frac{\mathrm dy}{\mathrm dx} = f(x)g(y)$$ + +We can solve these by rearranging: + +$$\frac1{g(y} \cdot \frac{\mathrm dy}{\mathrm dx} = f(x)$$ + +$$\int\! \frac1{g(y)} \,\mathrm dy = \int\! f(x) \,\mathrm dx + c$$ + + +
+ + +#### Example 1 + +Find $y$ such that + +$$\frac{\mathrm dy}{\mathrm dx} = ky$$ + +where $k$ is a constant. + + + +Rearrange to get + +\begin{align*} +\int\! \frac1y \,\mathrm dy &= \int\! k \mathrm dx + c \\ +\ln y &= kx + c +y &= e^{kx + c} = e^ce^{kx} \\ + &= Ae^{kx} +\end{align*} + +where $A = e^c$ is an arbitrary constant. + +
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