Integrals Problem

Question

Given that 7≤f(x)≤10 for 1≤x≤4, estimate the value of ∫₁⁴ f(x)dx

....... ≤ ∫₁⁴ f(x)dx ≤ ......

Answer 1

21 < integral < 30

y= 6 +x

Area under the line from x= 1 to 4 is

integral of 6x +x^2/2 evaluated at 1 to 4 = 24+4^2/2 - 6-1/2 = 32-6 1/2 = 25 1/2 as an rough estimate

but the area could range from (4-1)(7) = 21 to (4-1)10 = 30. That's the range of the integral.