Let f and g be functions that are continuous on the closed interval [1,5].
The definite integrals ∫15f(x)dx and ∫15g(x)dx are known to exist.
Answer the following questions about the integral of 7f(x)+g(x) over the same interval.
(a) Write an algebraic expression for ∫15(7f(x)+g(x))dx in terms of the two separate integrals ∫15f(x)dx and ∫15g(x)dx. No evaluation of numerical values is necessary.
(b) Provide a detailed justification, based on the interpretation of a definite integral as (signed) area, that supports the relationship you wrote in part (a). Your explanation should reference how areas under the graphs of f and g combine when the functions are scaled or added.