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Calculus/Area under a curve

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Expert: Alon Mandes - 12/10/2008


Question
a polynomial of degree 2 with a negative leading coefficiant has two distinct roots r1 and r2. Find the area of the closed region enclosed by the polynomial and the lines tangent to the curve at r1 and r2

Answer
Hello Corey, The general formula of a strait line is :
(y-y1)/(x-x1)=m. Where m is the slope, & (x1,y1) is the point
where the line passes through. In our case, m is the slope of our
function, meaning f'(x) at x=r1 or r2, & (x1,y1) is the point (r1,0)
or (r2,0). Hence the 2 tangents lines had the forms :
1. (y-0)/(x-r1)=f'(r1)
2. (y-0)/(x-r2)=f'(r2)
Our quadratic function is F=Ax^2+Bx+C, which gives F'(x)=2Ax+B.
& F'(r1)=2Ar1+B , F'(r2)=2Ar2+B.
plugging these results in equations (1) & (2) yields :
The tangent line at (r1,0) : y_left=(2Ar1+B)(x-r1)
The tangent line at (r2,0) : y_right=(2Ar2+B)(x-r2)
Now let's calculate area : The area R is :
∫ y_left - y_right - F(x) dx. Where x goes from r1 to r2.
∫ (2Ar1+B)(x-r1) - (2Ar2+B)(x-r2) - Ax^2+Bx+C  dx =
∫ 2A[r1-r2]x-2A[r1^2-r2^2]-B[r1-r2]-Ax^2-Bx-C.

I'll leave this immediate integral to you as an exercise.

Alon.  

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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