Find area between two curves.

Question

x^3/4 + y^3/4 = 1 

 

x+y=1 

Answer 1

Like Richard said, there is no closed form in terms of elementary functions for the antiderivative, so for first year calculus, you will have to approximate it on a calculator.

 

However, we can give it in terms of the gamma function:

 

Γ(z) = ∫₀^∞ t^z-1 e^-t dt for Re(z)>0, Γ(z+1)=zΓ(z) in the whole plane.

 

The Gamma function is a generalization of the factorial as Γ(n+1) = n! for all integers n≥0.

 

The exact area is A =1/2 - Γ(1/6)Γ(4/3) / (5·2^2/3·√π)

Answer 2

The two curves intersect at the points (0,1) and (1,0)

 Also,  x^3/4 + y^3/4 is less than x +y   for all points in the first quadrant.  

Thus the area between the curves is the integral over x  on the interval [0,1] of

  [ 1-x ] - [1 -x^3/4]^4/3

 

I don't think there is an antiderivative for this function.  

Numerical integration using a TI-84 calculator gave the result as 0.14667.