# In this worksheet, we will be finding the area between two curves using concepts learned in Calculus. 

# Sketch the graph of the curve f(x) = 6 - x ^ 2
plot( 6 - x ^ 2 , -10, 10 )

# On the same plane, sketch the curve above and f(x) = -10
plot ( 6 - x ^ 2, -10, 10 ) + plot ( -10 , -10, 10, color='red' ) 

# To find the area between the curves, we will have to determine where the curves intersect. These intersection points will become our interval for integration.
solve ( 6 - x ^ 2 == -10 , x )

# Now, we are going to integrate to find the area. 
# The formula for the area between to curves is int(upper)dx - int(lower)dx (bounded by the appropriate interval)
from sage.symbolic.integration.integral import definite_integral
definite_integral( 6 - x ^ 2 , x , -4 , 4 ) - definite_integral ( -10 , x , -4 , 4 )

# The solution to the definite integral, 256/3, is the area between the two curves. 



integral ( 6 - x ^ 2 , x ) - integral ( -10 , x)

# In my research for computing integrals in Sage, I found the computations below as a quick way to find all of the information we found above. I wanted to go through the steps one-by-one and then lay out the following as an alternative method.

b= 6 - x^2
r= -10
e=solve(b==r,x)
print "Points of intersection"
print e
d=integrate(b-r,-4,4)
print "Area under two curves is:"
print d