Josh Robbins
Vallieres Notes 11/4
Indefinate integral
restart;
f:= unapply(0.9*x^4- 2.4 * x^2- 1.5*x +0.2,x);
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int(f(x),x);
LCoqJEkieEc2IiIiJiQiKysrKys9ISM1KiRGJCIiJCQhKysrKyshKUYpKiRGJCIiIyQhKysrKyt2RilGJCQiKysrKys/Rik=
int(f(x),x=a..b);
LDIqJEkiYkc2IiIiJiQiKysrKys9ISM1KiRJImFHRiVGJiQhKysrKys9RikqJEYkIiIkJCErKysrKyEpRikqJEYrRi8kIisrKysrISlGKSokRiQiIiMkISsrKysrdkYpKiRGK0Y2JCIrKysrK3ZGKUYkJCIrKysrKz9GKUYrJCErKysrKz9GKQ==
Find the integral at the upper limit(a) kinus the integral of the lower limit(b). Find the area under the curve I guess. Integrals find areas!
Integral is an area and can be positive of negative
restart;
f:=unapply(0.9*x^4- 2.4 * x^2- 1.5*x +0.2,x);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiIlJCIiKiEiIiokRisiIiMkISNDRi9GKyQhIzpGLyRGMUYvIiIiRiVGJUYl
area:=unapply(int(f(x),x=a..b),a,b);
Zio2JEkiYUc2IkkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLDIqJDklIiImJCIrKysrKz0hIzUqJDkkRi0kISsrKysrPUYwKiRGLCIiJCQhKysrKyshKUYwKiRGMkY2JCIrKysrKyEpRjAqJEYsIiIjJCErKysrK3ZGMCokRjJGPSQiKysrKyt2RjBGLCQiKysrKys/RjBGMiQhKysrKys/RjBGJUYlRiU=
area(0,1.5);
JCErK11pP0YhIio=
area(k,m);
LDIqJEkibUc2IiIiJiQiKysrKys9ISM1KiRJImtHRiVGJiQhKysrKys9RikqJEYkIiIkJCErKysrKyEpRikqJEYrRi8kIisrKysrISlGKSokRiQiIiMkISsrKysrdkYpKiRGK0Y2JCIrKysrK3ZGKUYkJCIrKysrKz9GKUYrJCErKysrKz9GKQ==
plot(f(x), x=-2..2.5, title="f(x)");
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plot(area(-2,x),x=-2..2.5, title="area");
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Connection between definate and indefinate integral
restart;
f:=unapply(0.9*x^4- 2.4 * x^2- 1.5*x +0.2,x);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiIlJCIiKiEiIiokRisiIiMkISNDRi9GKyQhIzpGLyRGMUYvIiIiRiVGJUYl
F:=unapply(int(f(x),x),x);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiImJCIrKysrKz0hIzUqJEYrIiIkJCErKysrKyEpRi8qJEYrIiIjJCErKysrK3ZGL0YrJCIrKysrKz9GL0YlRiVGJQ==
area:=unapply(int(f(x),x=a..b),x) ;
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLDIqJEkiYkdGJSIiJiQiKysrKys9ISM1KiRJImFHRiVGLCQhKysrKys9Ri8qJEYrIiIkJCErKysrKyEpRi8qJEYxRjUkIisrKysrISlGLyokRisiIiMkISsrKysrdkYvKiRGMUY8JCIrKysrK3ZGL0YrJCIrKysrKz9GL0YxJCErKysrKz9GL0YlRiVGJQ==
`Area=indefinate integral at b- indefinate integral at a`;
area(a,b);
F(b)-F(a);
SVhBcmVhPWluZGVmaW5hdGV+aW50ZWdyYWx+YXR+Yi1+aW5kZWZpbmF0ZX5pbnRlZ3JhbH5hdH5hRzYi
LDIqJEkiYkc2IiIiJiQiKysrKys9ISM1KiRJImFHRiVGJiQhKysrKys9RikqJEYkIiIkJCErKysrKyEpRikqJEYrRi8kIisrKysrISlGKSokRiQiIiMkISsrKysrdkYpKiRGK0Y2JCIrKysrK3ZGKUYkJCIrKysrKz9GKUYrJCErKysrKz9GKQ==
LDIqJEkiYkc2IiIiJiQiKysrKys9ISM1KiRJImFHRiVGJiQhKysrKys9RikqJEYkIiIkJCErKysrKyEpRikqJEYrRi8kIisrKysrISlGKSokRiQiIiMkISsrKysrdkYpKiRGK0Y2JCIrKysrK3ZGKUYkJCIrKysrKz9GKUYrJCErKysrKz9GKQ==
`recoup f(x) by differentiationg F(x)`;
unapply(diff(F(x),x));
SUVyZWNvdXB+Zih4KX5ieX5kaWZmZXJlbnRpYXRpb25nfkYoeClHNiI=
Zio2IkYjNiRJKW9wZXJhdG9yR0YjSSZhcnJvd0dGI0YjLCoqJEkieEdGIyIiJSQiKysrKyshKiEjNSokRikiIiMkISsrKysrQyEiKkYpJCErKysrKzpGMiQiKysrKys/Ri0iIiJGI0YjRiM=
Fa-Fb is manually doing what the area funtion did
Total Charge on a wire
restart;
charge:=unapply(int(rho(x), x=a..b),a,b);
Zio2JEkiYUc2IkkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtSSRyaG9HRiU2I0kieEdGJS9GMzs5JDklRiVGJUYl
rho:=unapply(x*exp(-(x-0.5)**2),x);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJCwmRipGKyQhIiYhIiJGKyIiI0Y3RitGJUYlRiU=
plot(rho(x), x=-2..3, title="Charge Density");
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
charge(-2,3);
charge(1,2);
charge(-1.5,-.5);
charge(-1,.5);
charge(-infinity, infinity);
JCIrT0ZtZSkpISM1
JCIrSFphVGAhIzU=
JCErQUZgcjUhIzU=
JCErIjQ+MSM+ISM2
JCIrYiNwQScpKSEjNQ==
Chareg distribution on a plane
restart;
charge:=unapply(int(int(rho(x,y), y=c..d), x=a..b),a,b,c,d);
Zio2JkkiYUc2IkkiYkdGJUkiY0dGJUkiZEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtRi02JC1JJHJob0dGJTYkSSJ4R0YlSSJ5R0YlL0Y4OzkmOScvRjc7OSQ5JUYlRiVGJQ==
rho:=unapply(bump(r(x,y)),x,y);
LUkiQEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSVidW1wR0YnZio2JEkieEdGJ0kieUdGJ0YnNiRJKW9wZXJhdG9yR0YnSSZhcnJvd0dGJ0YnLUkickdGJzYkOSQ5JUYnRidGJw==
bump:=unapply(x*exp(-(x-0.5)^2),x);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJCwmRipGKyQhIiYhIiJGKyIiI0Y3RitGJUYlRiU=
r:=unapply(sqrt(x**2-y**2),x,y);;
Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsJiokOSQiIiMiIiIqJDklRi4hIiIjRi9GLkYlRiVGJQ==
plot3d(rho(x,y), x=-3..3,y=-3..3);
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plot3d(0.0, x=-3..3, y=-3..3,color=rho(x,y));
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