Let and be functions that are continuous on the closed interval

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let ff and gg be functions that are continuous on the closed interval [1,5][1,5]. The definite integrals ∫15f(x) dx\displaystyle \int_{1}^{5} f(x)\,dx and ∫15g(x) dx\displaystyle \int_{1}^{5} g(x)\,dx are known to exist. Answer the following questions about the integral of 7f(x)+g(x)7f(x)+g(x) over the same interval. (a) Write an algebraic expression for ∫15 ⁣(7f(x)+g(x)) dx\displaystyle \int_{1}^{5}\!\bigl(7f(x)+g(x)\bigr)\,dx in terms of the two separate integrals ∫15f(x) dx\int_{1}^{5} f(x)\,dx and ∫15g(x) dx\int_{1}^{5} g(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 ff and gg combine when the functions are scaled or added.

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