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Copy of Calculus II Review

Author:sonia7nis, ssvestal, Rasha Tarek

Surface of Revolution

In this section, you will create a surface of revolution by revolving a function around the -axis. In the Input bar below, graph f(x) = sin(x). Now to restrict the graph to the interval [0, ], put a comma after the function and then . Now we want to rotate this curve around the -axis in three dimensions. To do this, we need another variable, n, so simply type n into the input bar. Reset the slider so that it goes from 0 to 2. To rotate the curve and create a surface of revolution, type Surface(f,n,xAxis).

Solid of Revolution

In order to see a solid of revolution that is revolved around an axis, we must use an applet that was created by another Geogebra user. In the applet below, input as the upper function, as the lower function, rotate about the x-axis, from to . Then revolve the entire region. Basically, it will look like the curve from the previous problem, but now it is a solid of revolution.

Finding the volume of the solid of revolution.

For the solid that you just graphed, what method would be best to use to find the volume of the solid.

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Revolving around the y-axis

Taking the same upper function, lower function, initial point, and end point, now revolve the region around the y-axis. After examining this new solid of revolution, which method would be best to use to find the volume.

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Length of a parametric curve

Graph the curve in the graphing window below, using the Curve command. Once the curve is graphed, use the length command to find the length of the parametric curve.

Graphing in Polar Coordinates

In order to graph a curve defined in polar coordinates, there is a special way that we have to use the Curve command. Here is what you need to do: Curve((; ), , 0, 2). Using this special command, graph both polar curves in the graphing window below. Now, if we want to find where the two polar curves intersect, we can use the Intersect command. Type Intersect(a,b). You will notice that it only gives you the coordinates of one of the two points, and the coordinates are given in Cartesian coordinates.

Intersection of the two polar curves

As mentioned above, there are two angles where the polar curves intersect. Use your knowledge of algebra and trigonometry to find both angles. Then type the angles in the box below. You may write out pi instead of using the Greek letter.

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Taylor polynomials

Geogebra is helpful when finding the Taylor polynomial of a function, centered at a, of degree n. Remember that you simply type TaylorPolynomial[function,a,n]. In the input bar of the graph below, find the 5th degree Taylor polynomial for centered at .

Maclaurin polynomial

Geogebra only has the TaylorPolynomial command. What would one have to do in order for Geogebra to compute the Maclaurin polynomial for a function?

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Thoughts on Geogebra

As you are learning more Geogebra, do you think that it would be useful for instructors to include some of it in Calculus I and II?

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