Calculus

Calculus II: Project III Due date: November 07, 2018 at 10:00a.m 25 points

Directions: In this project a computer algebra system (mainly Mathematica) is used to investigate

integrals of certain parametric equations. You will turn in all the commands (formulas used) in

Mathematica and the results in a printed document, include cover page and no hand written part should

be included in a project. Thank you.

Tools_________________________________________________________________________________

Define functions: f[x_]:=

Graph parametric equations, with parameter interval [a,b]:

ParametricPlot[{𝑓1[𝑢], 𝑓2[𝑢]}, {𝑢, 𝑎, 𝑏}]

Graph several parametric equations, with parameter interval [a, b]:

ParametricPlot[{{𝑓1[𝑢], 𝑓2[𝑢]}, {𝑔1[𝑢], 𝑔2[𝑢]}, … }, {𝑢, 𝑎, 𝑏}]

To find the derivative: f’[x]

To integrate from a to b: Integrate[f[u],{u,a,b}]

To get a numerical approximation of the last output: N[%]

Remember, we enter commands by pressing SHIFT + ENTER

If you need other commands, consult the Index of Functions found under the Help menu.

_____________________________________________________________________________________

Part One.

(a) Graph the epitrochoid with parametric equations: 𝑥 = 11 cos 𝑢 − 4 cos(

11𝑢

2 )

𝑦 = 11 sin 𝑢 − 4 sin( 11𝑢

2 )

Find the parameter interval that gives the complete curve.

(b) Find the approximate length of this curve.

Part Two.

The Bezier curves are used in computer-aided design and are named after the French

mathematician Pierre Bezier (1910-1999), who worked in the automotive industry. A cubic Bezier curve

is determined by four control points, 𝑃0(𝑥0, 𝑦0), 𝑃1(𝑥1, 𝑦1), 𝑃2(𝑥2, 𝑦2), 𝑎𝑛𝑑 𝑃3(𝑥3, 𝑦3) and is defined by

the parametric equations:

𝑥 = 𝑥0(1 − 𝑢)

3 + 3𝑥1𝑢(1 − 𝑢) 2 + 3𝑥2𝑢

2(1 − 𝑢) + 𝑥3𝑢 3

𝑦 = 𝑦0(1 − 𝑢) 3 + 3𝑦1𝑢(1 − 𝑢)

2 + 3𝑦2𝑢 2(1 − 𝑢) + 𝑦3𝑢

3

where 0 ≤ 𝑢 ≤ 1. Notice that when u = 0 we have point 𝑃0 and when u = 1 we have point 𝑃3, so the

curve begins at 𝑃0 and ends at 𝑃3.

(a) Graph the Bezier curve with control points 𝑃0(4,1), 𝑃1(28,48), 𝑃2(50,42), 𝑎𝑛𝑑 𝑃3(40,5).

The line segment that joins points 𝑃1(𝑥1, 𝑦1) and 𝑃2(𝑥2, 𝑦2) is given by the parametric equations:

𝑥 = 𝑥1 + (𝑥2 − 𝑥1)𝑢

𝑦 = 𝑦1 + (𝑦2 − 𝑦1)𝑢

where 0 ≤ 𝑢 ≤ 1.

(b) Use this fact to graph the Bezier curve from part (a), along with the line segments𝑃0𝑃1, 𝑃1𝑃2,

𝑎𝑛𝑑 𝑃2𝑃3 , all on the same set of axes.

Notice that the middle control points 𝑃1 and 𝑃2 do not lie on the curve. The curve starts at 𝑃0,

heads towards 𝑃1 and 𝑃2 without reaching them, and ends at 𝑃3.

(c) Try to produce a Bezier curve that has a loop in it by changing 𝑃1 in part (a).

(d) Some laser printers use Bezier curves to represent letters and symbols. Experiment with

different sets of control points until you find a Bezier curve that gives a good representation of

the letter C.