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Expert: Sherry Wallin - 3/15/2011


Question
Determine the area bounded between the line y = | x – 1 | and circle x2 + y2 = 5

Answer

integration
Hi Vipul~

If you graph both in the first quadrant you will see that y = |x - 1| is 'above' x^2 + y^2 = 5 so you will need to find out where the two graphs meet so that you can find your limits of integration. First solve x^2 + y^2 = 5 for y since you already have |x - 1| = y so you will integrate both with respect to x.

x^2 + y^2 = 5 => y^2 = -x^2 + 5 or y = +- sqrt(5-x^2). Since we are only concerned with the positive sqrt or when y >= 0 we will use just y = sqrt(5-x^2).

Integrate notation: Int (a,b,y,x) means to integrate from a to b, limits on y with respect to the variable x.

Both functions need to be integrated from (0,1) with respect to x.
Thus Int(0,1){[|x-1|-(sqrt(5-x^2))]dx} = Int(0,1)[|x-1|dx] - Int(0,1)[sqrt(5-x^2)dx]. When you integrate |x-1| you can integrate x - 1 but just remember you want the positive value of whatever you get. You will find that when you integrate x - 1 you will get -1/2 so it's area under the curve is actually |-1/2| = 1/2.

You do the rest. Hope this helps.

Math Prof

Oh, and don't forget whatever value you get from this whole integration double it because to the left of the y-axis you get a mirror image and thus an equal area between the two curves.

I have also attached the result for integrating (5-x^2)^(1/2)  

Sherry Wallin

Expertise

I can answer most questions up through Calculus and some in Number Theory and Abstract Algebra.

Experience

I have had my Bachelor's Degree since 1987 and have been a teacher since 1988. I earned my Masters Degree in Mathematics May 2010. I have been teaching at the same community college since 2002.

Education/Credentials
I have taught 12 years at the community college level, medical college, and technical college as well as a high school instructor and alternative education instructor and charter school instructor.

Awards and Honors
Master's GPA 3.56 Bachelor's GPA 3.34 Post grad work not degree related GPA 4.0

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