MATH 156 LAB 4

In this lab we start with the observation that NiMvLSUkaW50RzYkLSUlc3FydEc2IywmIiIiRisqJCUieEciIiMhIiIvRi07LCRGK0YvRisqJiUjUGlHRitGLkYv .

This is so, because the integrand is the equation of a circle:

If we square y=NiMtJSVzcXJ0RzYjLCYiIiJGJyokJSJ4RyIiIyEiIg==,

we get NiMvKiQlInlHIiIjLCYiIiJGKCokJSJ4R0YmISIi,

which gives NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoLSUmRmxvYXRHNiRGKCIiIQ==

Let us graph the integrand:

f:=x->sqrt(1-x^2);

plot(f(x), x=-1..1);

This does not look like a (semi)circle. The reason is that Maple arranges the size of the axes to be pleasing to the eye, not to be "mathematically" correct.

To remedy the situation, we have to introduce the command "scaling=constrained", which tells Maple to use the same scale on the NiMlInhH and on the NiMlInlH axes.

This way there will be no distortion.

plot(f(x), x=-1..1, scaling=constrained);

If you want to see the lower semicircle, we have to introduce the function NiMvLSUiZ0c2IyUieEcsJC0lJXNxcnRHNiMsJiIiIkYtKiRGJyIiIyEiIkYw.

Introduce a command that defines this function.

We would not only like to plot this function, but also see the two plots together. Introduce commands that plot NiMtJSJnRzYjJSJ4Rw==, name this plot and the plot for NiMtJSJmRzYjJSJ4Rw== and display the two plots simultaneously. Do not forget to introduce the plots package.

This process is rather cumbersome. Maple has another command that will plot for us the circle with equation NiMvLCYqJCUieEciIiMiIiIqJCUieUdGJ0YoRig= in an easier way:

This is the implicitplot command:

implicitplot(x^2+y^2=1, x=-1..1, y=-1..1, scaling=constrained);

Notice that we also gave the range of y-values.

Plot the equation NiMvLCYqJCwmJSJ4RyIiIkYoISIiIiIjRigqJCwmJSJ5R0YoRipGKEYqRigiIiU=. How do you decide on the range of x and y-values?

This is the equation of a circle with radius 2 and center (1, -2). In general the equation of a circle with radius R and center at (a, b) is

NiMvLCYqJCwmJSJ4RyIiIiUiYUchIiIiIiNGKCokLCYlInlHRiglImJHRipGK0YoKSUiUkctJSZGbG9hdEc2JEYrIiIh.

The first plot we saw today looked like an elongated circle. This has the name ellipse in mathematics. An ellipse centered at the origin has an equation of the form

NiMvLCYqJiUieEciIiMqJCUiYUdGJyEiIiIiIiomJSJ5R0YnKiQlImJHRidGKkYrRis=. The NiMlInhH-intercepts of this equation are (NiQlImFHIiIh), (NiQsJCUiYUchIiIiIiE=). The NiMlInlH-intercepts of the ellipse are (NiQiIiElImJH), (NiQiIiEsJCUiYkchIiI=). The numbers NiMlImFH and NiMlImJH are called the lenghts of the semiaxes. The largest of the two semiaxes is the major semiaxis and the smallest is the minor semiaxis.

Let us plot two ellipses with equations: NiMvLCYqJiUieEciIiMiI0QhIiIiIiIqJiUieUdGJyIjO0YpRipGKg== and NiMvLCYqJiUieEciIiMiIiohIiIiIiIqJiUieUdGJyIjO0YpRipGKg==.

implicitplot(x^2/25+y^2/16=1, x=-5..5, y=-4..4, scaling=constrained);

implicitplot(x^2/9+y^2/16=1, x=-3..3, y=-4..4, scaling=constrained);

How do you decide on the range on the NiMlInhH and NiMlInlH values? What do you notice?

If you wonder why ellipses are important, the answer comes from astronomy: The trajectories of planets and satellites are ellipses. If the center of the ellipse is at the point (NiQlImNHJSJkRw==) and the semiaxes have length NiQlImFHJSJiRw== along the NiQlInhHJSJ5Rw== axes, then the equation of the ellipse is: NiMvLCYqJiwmJSJ4RyIiIiUiY0chIiIiIiMqJCUiYUdGK0YqRigqJiwmJSJ5R0YoJSJkR0YqRisqJCUiYkdGK0YqRihGKA==. Plot the ellipse with equation

NiMvLCYqJiwmJSJ4RyIiIiIiIyEiIkYpIiM7RipGKComLCYlInlHRihGKEYqRikiIipGKkYoRig=.

Sometimes we are given other equations that represent circles and ellipses but it is a bit harder to see what the geometric data of the plot are.

For instance, the ellipse you have just plotted can be given by the equation: NiMvLCoqJiIiKiIiIiokJSJ4RyIiI0YnRicqJiIjO0YnKiQlInlHRipGJ0YnKiYiI09GJ0YpRichIiIqJiIjS0YnRi5GJ0YxIiMjKg==. You can verify this by using the simplify command:

simplify((x-2)^2/16+(y-1)^2/9);

This just expands the left-hand side. To clear the denominators, we multiply with 16 X 9=144.

%*144;

It is easy to check that NiMvLCYiJFciIiIiIiNfISIiIiMjKg==.

144-52;

It would be nice if we could start from the equation NiMvLCoqJiIiKiIiIiokJSJ4RyIiI0YnRicqJiIjO0YnKiQlInlHRipGJ0YnKiYiI09GJ0YpRichIiIqJiIjS0YnRi5GJ0YxIiMjKg== and see the equation of the ellipse in the form NiMvLCYqJiwmJSJ4RyIiIiIiIyEiIkYpIiM7RipGKComLCYlInlHRihGKEYqRikiIipGKkYoRig=.

For this we need to complete the square in the equation. First of all we give a name to the left-hand side and introduce the student package:

left:=9*x^2-36*x+16*y^2-32*y; with(student):

newleft:=completesquare(left, {x, y});

The {NiQlInhHJSJ5Rw==} tells Maple to complete the square in both the NiQlInhHJSJ5Rw== variables. Now we move NiMsJCIjXyEiIg== to the right-hand side:

right:=52+92;

finalleft:=newleft+52;

finalleft/right;

eqn:=%=1;

Complete the square and identify the important geometric data for the equation NiMvLCoqJCUieEciIiMiIiIqJCUieUdGJ0YoKiYiIidGKEYmRighIiIqJiIiJUYoRipGKEYoIiIk. Plot the equation.

To verify the answer we also plot the original equation with the same ranges and we get the same plot:

implicitplot(x^2+y^2-6*x+4*y = 3, x=-1..7, y=-6..2, scaling=constrained);

Identify the equation NiMvLCgqJiIiKiIiIiokJSJ4RyIiI0YnRicqJiIiJUYnKiQlInlHRipGJ0YnKiYiI09GJ0YpRidGJyIiIQ==. Plot it directly and after completing the square.