Question

Let integral f(x)dx=9 from -6 to -1.5

integral f(x)dx=7 from -6 to -4.5

integral f(x)dx=4 from -3 to -1.5

A. Find integral f(x)dx from -4.5 to -3

B. Find integral (9f(x)-7)dx from from -4.5 to -3

Answer 1

you know that
∫[a,b] f(x) dx + ∫[b,c] f(x) dx = ∫[a,c] f(x) dx

That is adjoining intervals can be combined into one longer interval.

So,

∫[-6,-1.5] = ∫[-6,-4.5] + ∫[-4.5,-3] + ∫[-3,-1.5]
or,
9 = 7 + ∫[-4.5,-3] + 4
∫[-4.5,-3] = -2
and so on.

For the second, recall that
∫(a*f(x) + ∫g(x)) dx
= a∫f(x) dx + ∫g(x) dx
so, we have

∫[-4.5,3] 9f(x)-7 dx
= 9∫[-4.5,-3] f(x) dx - ∫[-4.5,-3] 7 dx
= 9(-2) - 7(-3-(-4.5))
= -28.5