You are here:
Calculus/Calc
Advertisement
Question
Suppose R is the region in the plane enclosed by y=x^2 and y=4
a)Compute the perimeter P and the area A of R, and then compute the ration q=A/(P^2)
Note by squaring the perimeter the ratio becomes independent of the units chosen to measure the region.
b)Compute this ratio Q=A/(P^2) for these four regions: the region R, a square, a circle, and an equilateral triangle. Draw the figure in increasing order of Q
First, we need to note that the curves gives x² = 4 as the intersectoin points.
This means x = ±2, so in both cases, y=4.
The perimeter can be found by taking twice the length on one side and adding in 4 for the top.
I believe the length on one side is found by taking ∫√(x² + (x²)²)dx.
That works out to ∫√x²√(1+x²)dx = ∫x√(1+x²)dx.
Let u=1+x², so du = 2 dx, so (1/2)du = dx, and I think that becomes the integral of u/2,
which is u²/4. Evaluate this from 0 to 2, multiply the answer by 2, add in the 4 for the top,
and you have got the total perimeter P.
To find the area A, integrate x² from 2 downto -2.
Once this is done, A/P² can be computed.
b) For a square, s=side, so area A=s² and perimeter P=4s;
for a circle, area A=πr² and perimeter P=4π; and, finally,
for an equilateral triangle with side s, area A = (s²√3)/4 and perimeter P = 3r.
Given this, you can compute Q = A/P² for a circle, square, and triangle.
Related Articles
Answers by Expert:
Scotto
Expertise
Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.
Experience
Experience in the area: I have tutored students in all areas of mathematics for over 25 years.
Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors.
Awards and Honors: I have passed Actuarial tests 100, 110, and 135.
Publications
Maybe not a publication, but I have respond to well oveer 7,500 questions on the PC.
Well over 2,000 of them have been in calculus.
Education/Credentials
I aquired well over 40 hours of upper division courses. This was well over the number that were required.
I graduated with honors in both my BS and MS degree from Oregon State University.
I was allowed to jump into a few junior level courses my sophomore year.
Awards and Honors
I have been nominated as the expert of the month several times.
All of my scores right now are at least a 9.8 average (out of 10).
Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.
©2012 About.com, a part of The New York Times Company. All rights reserved.