integral properties

[digital isle]

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[burnt geyser]

-7→-9∫f(x)dx=-(-9→-7∫f(x)dx)
so -9→-7∫f(x)dx=2
∫⅓f(x)dx=⅓∫f(x)dx
-7→7∫f(x)dx=-9→7∫f(x)dx-(-9→-7∫f(x)dx)=5-2=3
-7→7∫f(x)dx=3
but we want ⅓∫f(x)dx, so we multiply that by ⅓ to get simply 1

for the first thing this calls back to the first fundamental theorem of Calculus

a→b∫f(x)dx=F(b)-F(a)
so if we did:
b→a∫f(x)dx=F(a)-F(b)
which happens to be -(F(b)-F(a))

for the second thing this applies to the scalar multiple rule that is within limits and sums, where if it is a constant and can be factored out, then it can be removed entirely out of the limit/sum

and finally for the final thing we use a little bit of geometry:
we are given the desired area and a little bit extra that we want to remove, the area between -9 and -7, so we can do this by subtracting the area of this individual part, which is represented by the integral -9→-7∫f(x)dx