︠abbc19bf-031d-4d29-ab2f-080c7da19b1ei︠
%html
<h1>Definite Integrals</h1>
<p>This notebook will help you with the following:</p>
<ul>
<li>Estimating area under curves using left and right endpoint approximations.</li>
<li>Finding area under postive curves using definite integrals, as well as finding diplacement from velocity.</li>
<li>Finding area between functions and the x axis.</li>
<li>Finding average value.</li>
</ul>

︡5c09705c-a3b1-406b-97e9-7acbb7a9ea8e︡{"html": "<h1>Definite Integrals</h1>\n<p>This notebook will help you with the following:</p>\n<ul>\n<li>Estimating area under curves using left and right endpoint approximations.</li>\n<li>Finding area under postive curves using definite integrals, as well as finding diplacement from velocity.</li>\n<li>Finding area between functions and the x axis.</li>\n<li>Finding average value.</li>\n</ul>"}︡
︠ad474c91-45ab-4195-878d-85f1d8b35a20︠
#auto
var('x,y,t')
︡4d74189e-bf8c-4255-8efe-f5baa08b4d10︡{"stdout": "(x, y, t)"}︡
︠1b05ea7f-8a40-404f-a6fc-bce21a1837b2i︠
%html
<h2>Approximating Area</h2>
<p>If you enter a function, a range of $x$ values from $a$ to $b$, and a number $n$ of rectangles you want to create, the code below will compute both left and right endpoint approximations to the directed area under $f$.&nbsp; I said directed because if $a&gt;b$, then you will get a negative, and if $f&lt;0$ then you will get a negative as well.</p>

︡d46806f9-7252-4175-b0f4-e76326810cad︡{"html": "<h2>Approximating Area</h2>\n<p>If you enter a function, a range of $x$ values from $a$ to $b$, and a number $n$ of rectangles you want to create, the code below will compute both left and right endpoint approximations to the directed area under $f$.&nbsp; I said directed because if $a&gt;b$, then you will get a negative, and if $f&lt;0$ then you will get a negative as well.</p>"}︡
︠3ce11ebe-a314-4f36-9ffd-f46c3dd7a76b︠
f(x)=4-x^2
a=0
b=2
n=4

dx=(b-a)/n


xvalues=[(a+k*dx) for k in range(0,n+1)]
yvalues=[f(c) for c in xvalues]
dAvalues=[f(c)*dx for c in xvalues]

p=plot(f,a,b)
for k in range(0,n+1):
    p+=point((xvalues[k],yvalues[k]),rgbcolor='black')
    p+=line([(xvalues[k],0),(xvalues[k],yvalues[k])],rgbcolor='black')
for k in range(1,n+1):
    p+=polygon([(xvalues[k]-dx,0),(xvalues[k]-dx,yvalues[k]),(xvalues[k],yvalues[k]),(xvalues[k],0)],rgbcolor=(1,0,0),alpha=.2)
    p+=polygon([(xvalues[k-1]+dx,0),(xvalues[k-1]+dx,yvalues[k-1]),(xvalues[k-1],yvalues[k-1]),(xvalues[k-1],0)],rgbcolor=(0,0,1),alpha=.2)

leftsum=sum([dAvalues[k] for k in range(0,n)])
rightsum=sum([dAvalues[k] for k in range(1,n+1)])

xvalues.insert(0,"x")
yvalues.insert(0,"y")
dAvalues.insert(0,"dA = y dx")

html.table([
xvalues,yvalues,dAvalues
])

p.show()

html.table([
["Left Sum",leftsum,leftsum.n()],
["Right Sum",rightsum,rightsum.n()],
["Average Sum",(leftsum+rightsum)/2,((leftsum+rightsum)/2).n()],
["Actual Area",integrate(f,a,b),integrate(f,a,b).n()]
])
︡b88ab58d-5574-4a0b-98f8-d556d18e1226︡{"html": "\n<div class=\"notruncate\">\n<table class=\"table_form\">\n<tbody>\n<tr class =\"row-a\">\n<td>x</td>\n<td><span class=\"math\">0</span></td>\n<td><span class=\"math\">\\frac{1}{2}</span></td>\n<td><span class=\"math\">1</span></td>\n<td><span class=\"math\">\\frac{3}{2}</span></td>\n<td><span class=\"math\">2</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>y</td>\n<td><span class=\"math\">4</span></td>\n<td><span class=\"math\">\\frac{15}{4}</span></td>\n<td><span class=\"math\">3</span></td>\n<td><span class=\"math\">\\frac{7}{4}</span></td>\n<td><span class=\"math\">0</span></td>\n</tr>\n<tr class =\"row-a\">\n<td>dA = y dx</td>\n<td><span class=\"math\">2</span></td>\n<td><span class=\"math\">\\frac{15}{8}</span></td>\n<td><span class=\"math\">\\frac{3}{2}</span></td>\n<td><span class=\"math\">\\frac{7}{8}</span></td>\n<td><span class=\"math\">0</span></td>\n</tr>\n</tbody>\n</table>\n</div>\n"}︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡{"html": "\n<div class=\"notruncate\">\n<table class=\"table_form\">\n<tbody>\n<tr class =\"row-a\">\n<td>Left Sum</td>\n<td><span class=\"math\">\\frac{25}{4}</span></td>\n<td><span class=\"math\">6.25000000000000</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>Right Sum</td>\n<td><span class=\"math\">\\frac{17}{4}</span></td>\n<td><span class=\"math\">4.25000000000000</span></td>\n</tr>\n<tr class =\"row-a\">\n<td>Average Sum</td>\n<td><span class=\"math\">\\frac{21}{4}</span></td>\n<td><span class=\"math\">5.25000000000000</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>Actual Area</td>\n<td><span class=\"math\">\\frac{16}{3}</span></td>\n<td><span class=\"math\">5.33333333333333</span></td>\n</tr>\n</tbody>\n</table>\n</div>\n"}︡
︠e8eaf2c4-107c-4738-83e9-e3e6e4c76460i︠
%html
<h2>Definite Integrals $\displaystyle \int_a^b f(x) dx$</h2>
<p>The following code will compute definite integrals, as well as graph a function and shade the region between the function and the $x$-axis.</p>
<ul>
<li>If you type in velocity, it will find the change in position. </li>
<li>&nbsp;If you type in a positive function, it will compute the area under the graph. </li>
</ul>
<p>You can type in your functions using $t$, $x$, or any other variable you wish.</p>

︡e96a0b3d-bb5c-4ce9-99a2-4f87fa15d3e9︡{"html": "<h2>Definite Integrals $\\displaystyle \\int_a^b f(x) dx$</h2>\n<p>The following code will compute definite integrals, as well as graph a function and shade the region between the function and the $x$-axis.</p>\n<ul>\n<li>If you type in velocity, it will find the change in position. </li>\n<li>&nbsp;If you type in a positive function, it will compute the area under the graph. </li>\n</ul>\n<p>You can type in your functions using $t$, $x$, or any other variable you wish.</p>"}︡
︠d9727ca0-1054-4f99-a5da-58ff6780e291︠
v(t)= 4-t^2
a=0
b=5

p=plot(v,a,b,fill=true).show()
integrate(v,a,b)
︡5a8cebfb-d8aa-44bf-80c5-fe3f13e4ae83︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡{"stdout": "-65/3"}︡
︠ed615ee4-61b7-4128-9313-0e0267cb0639i︠
%html
<h2>Area between $f$ and the $x$-axis.</h2>
<p>If you want to find the area between $f$ and the $x$-axis, the first step is to find the zeros of the function. Then you break the integral to integrate over positive and negative parts separately.&nbsp; The code below will find the zeros, give you the integral, tell you the area from both the positive and negative parts, as well as the total area.</p>

︡e0596da8-243e-4e50-9c96-0e2afc64f260︡{"html": "<h2>Area between $f$ and the $x$-axis.</h2>\n<p>If you want to find the area between $f$ and the $x$-axis, the first step is to find the zeros of the function. Then you break the integral to integrate over positive and negative parts separately.&nbsp; The code below will find the zeros, give you the integral, tell you the area from both the positive and negative parts, as well as the total area.</p>"}︡
︠5af96e10-4f68-4b0a-8be4-1ec1caa6ec9d︠
f(x) = x^2-x-6
a=-5
b=5

x_intercepts = solve(f,x)
p=plot(f,a,b,fill=true)
DI=integrate(abs(f(x)),a,b)
g=abs(f)
Area=g.nintegrate(x,a,b)
PA=((g+f)/2).nintegrate(x,a,b)
NA=((g-f)/2).nintegrate(x,a,b)


html.table([
["x intercepts", x_intercepts],
["Integral of f", integrate(f(x),x)],
["Total Area", Area[0]],
["Positive Area", PA[0]],
["Negative Area", NA[0]],
])
p.show()
︡3b896fac-ba81-463b-8522-bb343ae96ff7︡{"html": "\n<div class=\"notruncate\">\n<table class=\"table_form\">\n<tbody>\n<tr class =\"row-a\">\n<td>x intercepts</td>\n<td><span class=\"math\">\\text{[\nx == 3,\nx == -2\n]}</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>Integral of f</td>\n<td><span class=\"math\">\\frac{1}{3} \\, x^{3} - \\frac{1}{2} \\, x^{2} - 6 \\, x</span></td>\n</tr>\n<tr class =\"row-a\">\n<td>Total Area</td>\n<td><span class=\"math\">65.0</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>Positive Area</td>\n<td><span class=\"math\">44.1666666667</span></td>\n</tr>\n<tr class =\"row-a\">\n<td>Negative Area</td>\n<td><span class=\"math\">20.8333333333</span></td>\n</tr>\n</tbody>\n</table>\n</div>\n"}︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡
︠14f87079-ed25-4211-aed6-d5b8f2ff370di︠
%html
<h2>Average Value $\displaystyle \frac{1}{b-a}\int_a^b f(x)\ dx$</h2>
<p>The code below computes the average value of a function. Remember that the average value of a continuous function always passes through the function.&nbsp; The average value $AV$ is the $y$ value so that the area above $AV$ below $f$ is the same as the area below $AV$ and above $f$. I like to think of average value as the height of sand in an ant farm that would result from shaking the ant farm to make a level top.</p>

︡207698dc-a51c-46d7-9915-29e5c0485929︡{"html": "<h2>Average Value $\\displaystyle \\frac{1}{b-a}\\int_a^b f(x)\\ dx$</h2>\n<p>The code below computes the average value of a function. Remember that the average value of a continuous function always passes through the function.&nbsp; The average value $AV$ is the $y$ value so that the area above $AV$ below $f$ is the same as the area below $AV$ and above $f$. I like to think of average value as the height of sand in an ant farm that would result from shaking the ant farm to make a level top.</p>"}︡
︠602360f5-21ed-4454-8b89-d8c854646573︠
f(x) = 4-x^2
a=0
b=2

average_value = 1/(b-a)*integrate(f,a,b)
sols=solve(f(x)==average_value,x)

html.table([
["Average Value", average_value],
["x values", sols],
])
p=plot(f,a,b,fill=true,fillalpha=.2,fillcolor='red')
p+=plot(average_value,a,b,fill=true,fillalpha=.2,fillcolor='blue',color='black')
p.show()
html.table([["Notice how the area above the average value is the same as the area below."]])
︡d9b4b0f7-ea6a-486a-8d9b-9ed62472b528︡{"html": "\n<div class=\"notruncate\">\n<table class=\"table_form\">\n<tbody>\n<tr class =\"row-a\">\n<td>Average Value</td>\n<td><span class=\"math\">\\frac{8}{3}</span></td>\n</tr>\n<tr class =\"row-b\">\n<td>x values</td>\n<td><span class=\"math\">\\text{[\nx == -2/3*sqrt(3),\nx == 2/3*sqrt(3)\n]}</span></td>\n</tr>\n</tbody>\n</table>\n</div>\n"}︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡{"html": "\n<div class=\"notruncate\">\n<table class=\"table_form\">\n<tbody>\n<tr class =\"row-a\">\n<td>Notice how the area above the average value is the same as the area below.</td>\n</tr>\n</tbody>\n</table>\n</div>\n"}︡