"Testing defint"

clear

T = defint(x^2,y,0,sqrt(1-x^2),x,-1,1)
check(T = 1/8*pi)

f = (4 x^2 + 5)^(1/2)
T = defint(f,x,0,1,y,0,1) -- uses integral #138
check(T = 3/2 + 5/8 log(5))

f = (u^2 + 1)^(1/2)
T = defint(f,u,0,1,v,0,3pi)
check(T = 3/2*pi*log(1+2^(1/2))+3*pi/(2^(1/2)))

f = -2 x^4 y^2 + x^4 y - 4 x^3 y - 2 x^2 y^4 + 2 x^2 y^3 + 2 x^2 y^2 - 2 x^2 y + y^5 - 2 y^3 + y
T = defint(f,y,-sqrt(1-x^2),sqrt(1-x^2),x,-1,1)
check(T == 1/48 pi)

f = 3 r^3 cos(theta)^2 + 3 r^3 sin(theta)^2
T = defint(f,r,0,1,theta,0,2pi)
check(T == 3/2 pi)

-- verify that defint works in a denominator

clear

I(y) = defint((1 + cos(theta)^2) sin(theta),theta,0,y)
A = I(theta) / I(pi)
check(A = -1/8 cos(theta)^3 - 3/8 cos(theta) + 1/2)
