Above the x-axis
If #f(x)# is non-negative over the interval #[a,b]#, then
#int_a^b f(x) dx# is the area of the region between the graph and the #x#-axis between #x=a# and #x=b#
Below the x-axis
If #f(x)# is non-positive over the interval #[a,b]#, then
#int_a^b f(x) dx# is the negative of area of the region between the graph and the #x#-axis between #x=a# and #x=b#
Example:
#int_0^3 sqrt(9-x^2) dx#.
#a=0# and #b=3#.
#f(x) = sqrt(9-x^2)# is never negative, so it is not negative on #[0,3]#
Therefore the integral is equal to the area under the curve and above the #x#-axis between #x=0# and #x=3#.
The graph of #y = sqrt(9-x^2)# is the part of #y^2 = 9-x^2# that has non-negative #y#-values. It is the upper semicircle for #x^2+y^2 = 9#
The part between #x=0# and #x=3# is a quarter of a circle with radius 3.
graph{y = sqrt(9-x^2)*(sqrt(1.5^2-(x-1.5)^2))/(sqrt(1.5^2-(x-1.5)^2)) [-2.63, 6.137, -0.812, 3.572]}
So
#int_0^3 sqrt(9-x^2) dx# is #1/4# of the area of the circle with radius #3#
#int_0^3 sqrt(9-x^2) dx = (9 pi)/4#
