Expert: Abe Mantell - 7/17/2011

Question
Hello ,
I am having difficulties trying to get the value of q. I have look for examples in my book for similar questions but there are none.

My question is given two functions: H(x) and P(x),
H(x)=x^(2)   and  P(x) = 4-x^(2 )- q*x.
Note, the function P also has a parameter, q which is a real number.


Find the real value(s) of the parameter q such that the area of the region enclosed between these two functions is equal to 25.

Thank you for your help
Monica

Answer
Hello Monica,

The area between the two is given by:
Integral[(4-x^2-qx) - x^2, x from x1 to x2], where x1 and x2 are where
the two curves intersect.  Solving x^2 = 4-x^2-qx gives
x1=-(1/4)*q-(1/4)*sqrt(q^2+32) and x2=-(1/4)*q+(1/4)*sqrt(q^2+32)

The algebra is very messy with those limits of integration, but works out as:
(1/24)*(q^2+32)^(3/2), now set this equal to 25 and solve for q...
q=(+ or -)2*sqrt(75^(2/3)-8) or about (+ or -) 6.256026382

OK?

Abe