Name:_____________________ Improper Integrals Done Properly



Math 2414 Activity 11 (Due by August 7)

1. Suppose that after a string is wound clockwise around a circle of radius a, its free end is at the point [pic]. Now the string is unwound, always stretched tight so the unwound portion [pic]is tangent to the circle at [pic]. The set of points traced out by the free end of the string is called the involute of the circle.

Find the parametric equations of the involute of the circle.

[pic]

2. Suppose the circle in the previous problem represents the cross-section of a cylindrical water tank of radius a, and the string is a rope of length [pic]. The rope is anchored at the point B opposite point A. If the other end of the rope is tied to a cow, let’s examine the region that can be grazed by the cow. Here is a diagram showing the rope in various positions:

[pic]

The boundary of the grazing region can be broken down into three pieces: [pic] is a portion of the involute of the circle, [pic] is a semicircle, and [pic] is the reflection across the x-axis of a portion of the involute.

Find the length of the boundary of the grazing region.

3. Find the area of the grazing region.

4. Now suppose that a sea cow(manatee) is tied to a point on the surface of a sphere of radius a by a rope of length [pic].

a) find the surface area of the grazing region of the sea cow.

b) Find the volume of the grazing region of the sea cow.

5. A 10 foot ladder slides down a wall as the bottom of the ladder is pulled away from the wall. Using the angle [pic] as a parameter, find the parametric equations for the path followed by

a) the top of the ladder, A. b) the bottom of the ladder, B.

[pic] [pic]

c) the point P on the ladder that is 4 feet from the top of the ladder.

[pic]

d) What familiar shape will be traced out by the point P as the ladder falls against the wall to the ground?[pic]

6. In the theory of the diffraction of light, the Spiral of Cornu occurs. It is given parametrically by [pic].

a) Find the length of the portion of the Spiral of Cornu corresponding to [pic].

b) Using the result from Activity #8 Problem #43, that [pic], find the limiting position of the Spiral of Cornu as [pic].

7. In computer-aided design special parametric curves called Bezier curves are used extensively. A cubic Bezier curve is determined by an ordered set of four points in the plane, called the control points of the curve. Given the four points [pic], [pic], [pic], and [pic], the cubic Bezier curve is defined by

[pic]

The curve begins at [pic], ends at [pic], and its shape is influenced by the positions of [pic] and [pic] For example, using the control points [pic], you get the Bezier curve:

Similarly, using the three sets of control points [pic] you can

draw a curve that approximates a letter.

Graph the Bezier curves that correspond to the given set(s) of control points:

a) [pic] b) [pic]

c) [pic]

8. In each of the following cases, find the four control points for the given Bezier curves:

a) The line: [pic] b) The parabola: [pic]

9. A very famous paradox, actually a fallacy, is the case of the concentric rolling disks. Two concentric disks, one disk of radius 1 and the other of radius 3 are fused together and the larger disk is rolled without slipping along the x-axis from 0 to [pic]. If you assume that the smaller disk will also roll without slipping along a parallel line segment 2 units above the x-axis, then you arrive at the apparent paradox of having the circumference of the smaller disk equal to the circumference of the larger disk.

Let’s find parametric equations for the two fixed points on the edge of the two disks a and b as the fused disks roll along the x-axis:

Assuming that the larger disk doesn’t slip, the point b will trace out a cycloid with the parametric equations: [pic]

And relative to the larger disk, the point a will trace out a curtate cycloid with parametric equations:[pic]

a) In this case of assuming that the larger disk doesn’t slip, find the lengths of the paths traveled by the points a and b. (Approximate, if necessary)

Assuming that the smaller disk doesn’t slip as it rolls along the segment 2 units above the x-axis, the point a would trace out a cycloid along this segment with parametric equations:

[pic]

And relative to the smaller disk, the point b would trace out a prolate cycloid with parametric equations:[pic]

b) In this case of assuming that the smaller disk doesn’t slip, find the lengths of the paths traveled by the points a and b. (Approximate, if necessary)

c) Explain the faulty assumption in this situation that leads to the apparent paradox.

10. Find the points on the parabola [pic] closest to the point [pic].

11. A cycloid is the curve generated by a point on a wheel as the wheel rolls. This diagram shows half of a loop of the standard cycloid, whose parametric equations are [pic]:

[pic]

Which is larger, the area of region I or the area of region II?

{Hint: The area of region I can be determined by subtraction of areas, and region II is just a square.}

12. Consider Maclaurin’s Trisectrix [pic].

a) Find the point on the curve where the curve crosses itself.

b) Find the slopes of the two tangent lines at the point of self crossing.

13. Find the tangent lines to the curve [pic] which pass through the origin.

14. Find the lines which are simultaneously a tangent line to the parabola [pic] for some parameter value s and a normal line to the parabola [pic] for some parameter value t.

15. Find the length of the cardioid [pic] for [pic].

16. Find the length of the curve [pic] for [pic].

17. Find the volume of the solid generated by revolving the region bounded by the curve [pic] (one arch of a cycloid) and the x-axis about the x-axis.

{Hint: Just replace the components of the volume formula [pic] by their parametric equivalents.}

18. A missile is fired from 500 miles away and follows a flight path given by [pic]. Two minutes later, an interceptor missile is fired which follows the flight path given by [pic]. Will the interceptor missile hit its target?

{Hint: In order for the interceptor missile to hit its target, they have to be at the same place at the same time.}

19. Find the area enclosed by the hypocycloid with parametric equations [pic].

20. What is the ratio of the area enclosed by the ellipse [pic] to the area of the largest rectangle that can be inscribed in the ellipse?

{Hint:

To find the area of the largest inscribed rectangle, you can maximize

[pic].

To find the area enclosed by the ellipse, you can use the parametrization:

[pic]}.

21. The parametric equations [pic] describe a portion of a parabola. Here’s why: [pic], so [pic].

To find the length of the parametric curve, we’d evaluate

[pic]

Letting [pic], we’d get

[pic]. Letting [pic], we’d get [pic]. Which means that the length of the parametric curve is

[pic]. From the graph, you can see that the length of the portion of the parabola must lie between the sum of the lengths of segments [pic] and [pic] and the sum of the lengths of segments [pic], [pic], and [pic]. [pic], and [pic]. So is it true that

[pic]? If not, explain why.

{Hint: How many times is the curve traced by the parametrization?}

22. Cooling towers for some power plants are made in the shape of a hyperboloid of one sheet.

This shape is chosen because it uses all straight reinforcing rods. A framework is constructed, then concrete is applied to form a relatively thin shell that is quite strong, yet has no structure inside to get in the way.

Suppose that the surface of the cooling tower is formed by revolving the hyperbola generated by the parametric equations [pic] about the y-axis, where x and y are in feet.

a) The bottom of the hyperboloid has y-coordinate of 0. What is the radius of the hyperboloid at its bottom?

b) What is the radius of the top of the hyperboloid?

c) What is the radius of the tower at its narrowest? How high up is the smallest radius?

d) Find the surface area of the hyperboloid.

e) If the walls of the tower are 4 inches thick, how many cubic yards of concrete will be needed to build the tower?

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Missile’s path

Interceptor missile’s path

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