MATH 4B -- LAB
2 -- HYPERBOLIC SINE AND COSINE NAME ________________
1. Set
up a definite integral to calculate the area of the region in the figure below (Figure
7.8.3 (b) in our textbook)...
i.e. the area
bounded by
a. The right half of the unit hyperbola x2
- y2
= 1
b. The x-axis
c. The
line from the origin to the point (cosh t, sinh t), which could be any point located on the unit
hyperbola in the first quadrant. This
can be written in the form y=mx, since the line
passes through the origin. Treating t as
a constant, what is the slope m of this line? ________________
How
many integrals would you need to set up the region as:
As a type I region (using vertical rectangles)? _______________________
As a type II region (using horizontal
rectangles)? ____________________
What
integral(s) did you use to represent the area?
______________________________________
2. Use
Maple to calculate the integral(s). The
result is an expression that represents the area.
[The area expression may
contain both the parameter t and either x or y (depending on whether you used a
type I or type II region). If so,
rewrite your area expression entirely in terms of t That is, replace x by cosh(t) or replace y by sinh(t).]
After my integral, and before the semicolon,
I had to type the words assuming real. I
then followed this with a simplify command:
f:= integrand;
g:=int(f,t=0..sinh(t))
assuming real); simplify(%);
Write your simplified integration result: ________________________________________________
________________________________________________________________________________
3.
Use Maple to graph the area (from step 2)
on the interval from t = 0 to t = 2. (Now t is treated as the
variable). I used plot(g,t=0..2);
4. You should have gotten a linear graph. Estimate its slope from the Maple graph: ______________
Based on the graph, write an equation that
expresses the apparent relationship between the area A and t
________________________________
5. Confirm your result from the textbook: See the discussion that accompanies Figure
7.8.3
6. Simplify the area expression from step 2
manually (paper-and-pencil -- this is not a computer step) to prove that your
conclusion from step 4 is correct.
7. Turn
in
a. This worksheet.
b. A printout of your
Maple work
c. Your
paper-and-pencil calculations for the setup of the integral
d. Your
paper-and-pencil calculations for the simplification in step 7
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Merced College; Don Power Updated 9/20/07 by Don Power