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Compute the area and volume of solids of revolution of e-2x

Define the function for all . Let

Compute

1. A(t);
2. V(t);
3. W(t);
4. .

1. The area of the ordinate set on is given by the integral,

2. The volume of the solid of revolution obtained by rotating about the -axis is

3. To compute the volume of the solid of revolution obtained by rotating about the -axis we first find as a function of .

Since , the integral is then from to 1 and we have

4. Finally, using parts (c) and (d) we can compute the limit,

Find a curve bisecting the curves x2 and x2/2

Consider the figure

We say the curve bisects the region between and in area if for every point on the curve the area of regions and are equal. Given the equations

find an equation for such that the curve bisects the region between and in area.

First, we can calculate the area of the region . This is the difference in the integrals from 0 to the -coordinate of of and . Since lies on the curve defined by the equation we may write for some . Then the area of is given by

Now, we make the assumption that the equation for is of the form for some positive real number . (I don’t know a good way to justify this assumption other than it’s the most obvious first thing to try, and it happens to work.) Then to find the area of region first we find equations for the curves and in terms of (so that we may integrate along the -axis which is somewhat easier). So we have

Then, we integrate along the -axis from 0 to (since this is the -coordinate of ) the difference between these two curves. The area of is then

Now we set the areas of the regions equal and solve for ,

Therefore, the equation describing is given by

Sketch a “limacon” and compute its area from 0 to 2 π

Define a limacon by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

Sketch a “cardioid” and compute its area from 0 to 2 π

Define a cardioid by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

where we know from this exercise (Section 2.11, Exercise #7).

Sketch a “four-leaf clover” and compute its area from 0 to 2 π

Define a four-leaf clover by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

where for the final step we used the previous exercise (Section 2.11, Exercise #12).

Sketch a “lazy eight” and compute its area from 0 to 2π

Define a lazy eight by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

Sketch a “four-leaved rose” and compute its area from 0 to 2π

Define a four-leaved rose by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

Sketch a “rose petal” and compute its area from 0 to π/2

Define a rose petal by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

Sketch two circles tangent to the x-axis and compute their area from 0 to 2π

Define two circles tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

Sketch a circle tangent to the x-axis and compute its area from 0 to π

Define a circle tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,