MAT 156 Lab 7 Topic 1: Area between two curves. We have seen how to graph two functions simultaneously. We have also seen how to solve equations. With these two processes, we can calculate the area between two curves. Example: Find the area between the parabola 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 and the line 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. f:=x->2*x^2;g:=x->-2*x+4;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwkKiYiIiMiIiIpJSJ4RyIiIyIiIiIiIjYiNiI2Ig== NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiYiIiMiIiIlInhHIiIiISIiIiIlIiIiNiI2IjYi

plot({f(x), g(x)}, x=-3..3);

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

By looking at the graph it seems rather obvious that the two curves intersect when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GKw== and when 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. We can verify that by setting the two equations to be equal and solving: fsolve(f(x)=g(x), x);

NiQkISsrKysrPyEiKiQiKysrKys1RiU=

For later purposes it will be interesting to give names to the two solutions. We achieve this by labeling the result of the previous command by sol: sol:=fsolve(f(x)=g(x), x);

NiM+JSRzb2xHNiQkISsrKysrPyEiKiQiKysrKys1ISIq

Now we can recover the two solutions as sol[1] and sol[2]: sol[1];

NiMkISsrKysrPyEiKg==

sol[2];

NiMkIisrKysrNSEiKg==

To find the area between the two curves we use the formula 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 where the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGNEY3Rj5GKw== is the top function and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ4RidGNEY3Rj5GKw== is the bottom function. Also LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-coordinate of the point furthest left and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-coordinate of the point furthest to the right on the region we study. Since in our example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEieEYnRjZGOUZALUY9Ni1RIjxGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGXG8tRiM2JS1GLDYlUSJnRidGNkY5RjxGVkYrRis= in the region we are interested we do: A:=Int(g(x)-f(x), x=sol[1]..sol[2]);

NiM+JSJBRy0lJEludEc2JCwoKiYiIiMiIiIlInhHIiIiISIiIiIlIiIiKiYiIiMiIiIpJSJ4RyIiIyIiIiEiIi8lInhHOyQhKysrKys/ISIqJCIrKysrKzUhIio=

value(A);

NiMkIiIqIiIh

Assignment Part 1: Find the area between the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtRiw2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNThGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GPVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkcvJSlzdHJldGNoeUdGRy8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVi1JKG1mZW5jZWRHRiQ2JS1GIzYjLUYsNiZRInhGJ0Y2RjlGPEY5RkMtRkA2LlEiPUYnRjlGQ0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjI3Nzc3NzhlbUYnL0ZYRl9vLUYjNiVGaG4tRkA2LlEiK0YnRjlGQ0ZFRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRmdvLUkjbW5HRiQ2JVEiMUYnRjlGQ0YrRis= and 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. Assignment Part 2: Find the area between the two curves LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYmUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUTJDaGFyYWN0ZXJTdHlsZTI2MUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0Y3L0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYjNiMtSSVtc3VwR0YkNiUtRiw2JlEieEYnRjRGN0Y6LUkjbW5HRiQ2JVEiNkYnRjdGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0Yr and 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.

Topic 2: Volumes of revolution. Example: Let R be the arc bounded above by the graph of f(x) = x^2-1 and below by the x-axis. Compute the volume of the solid obtained by rotating R about the x-axis. We can use the maple student calculus package with(Student[Calculus1]): f:=x->x^2-1;

NiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0c2IiwmKiQpJSJ4RyIiIyIiIiIiIiIiIiEiIjYiNiI2Ig==

sol:= fsolve(f(x)=0);

NiM+JSRzb2xHNiQkISsrKysrNSEiKiQiIiIiIiE=

V:=VolumeOfRevolution(f(x), x=sol[1]..sol[2], output=integral,axis=horizontal);

NiM+JSJWRy0lJEludEc2JComJSNQaUciIiIpLCYqJCklInhHIiIjIiIiIiIiIiIiISIiIiIjIiIiLyUieEc7JCErKysrKzUhIiokIiIiIiIh

value(V);

NiMkIitrQC5eTCEiKg==

Now compute the volume obtained by rotating about the line x=-2 V:=VolumeOfRevolution(f(x), x=sol[1]..sol[2], output=integral,axis=vertical,distancefromaxis=-2);

NiM+JSJWRy0lJEludEc2JCwkKioiIiMiIiIlI1BpRyIiIiwmJSJ4RyIiIiIiIyIiIiIiIiwmKiQpJSJ4RyIiIyIiIiIiIiIiIiEiIiIiIiEiIi8lInhHOyQhKysrKys1ISIqJCIiIiIiIQ==

value(V); To view the solid we can use Maple's plot3d command: the "t" parameter measures the angle of rotation:

NiMkIisjMztibiIhIik=

plot3d([x, f(x)*cos(t), f(x)*sin(t)], x=-1..1, t=0..2*Pi, axes=BOXED);

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Assignment Part 3: Let R be the region bounded above by the graph of 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 and below by the x-axis. A) Find the volume of the solid obtained by rotating R about the line x=-3 B) Find the volume of the solid obtained by rotating R about the y-axis. Assignment Part 4: Find the volume of the ellipsoid obtained by rotating the area inside the ellipse LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiUtSSZtZnJhY0dGJDYoLUYjNiMtSSVtc3VwR0YkNiUtRiw2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUyNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYlUSIyRidGQS9GRVEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYjNiMtRkg2JVEiOUYnRkFGSy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWi8lKWJldmVsbGVkR1EmZmFsc2VGJy1JI21vR0YkNi5RIitGJ0ZBRksvJSZmZW5jZUdGaW4vJSpzZXBhcmF0b3JHRmluLyUpc3RyZXRjaHlHRmluLyUqc3ltbWV0cmljR0Zpbi8lKGxhcmdlb3BHRmluLyUubW92YWJsZWxpbWl0c0dGaW4vJSdhY2NlbnRHRmluLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGXnAtRjQ2KC1GIzYjLUY5NiUtRiw2JlEieUYnRj5GQUZERkdGTS1GIzYjLUZINiVRIzQ5RidGQUZLRlVGWEZlbkZnbi1GW282LlEiPUYnRkFGS0Zeb0Zgb0Zib0Zkb0Zmb0Zob0Zqby9GXXBRLDAuMjc3Nzc3OGVtRicvRmBwRmNxLUZINiVRIjFGJ0ZBRktGKw== about the x-axis. The methods VolumeOfRevolution and plot3d can be used to display the three dimensional images. You can learn more about this by either reading the Maple Help files (accessible by typing ?plot3d or ?VolumeOfRevolution) or by looking at lab 7 on the department home page for 156.