Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use the values \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\) to evaluate the definite integral. (a) \(\int_{0}^{5}[f(x)+g(x)] d x\) (b) \(\int_{0}^{5}[f(x)-g(x)] d x\) (c) \(\int_{0}^{5}-4 f(x) d x\) (d) \(\int_{0}^{5}[f(x)-3 g(x)] d x\)

Step-by-Step Solution

Answer

The solutions to the integrals are: (a) 8, (b) 4, (c) -24, and (d) 0

Step 1: Evaluate the integral of the sum/difference of functions

To evaluate the integral of a sum or difference of functions like \(\int_{0}^{5}[f(x)\pm g(x)] d x\), use the property \(\int_a^b [f(x)\pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx\). Given that \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\), the definite integral becomes \(\int_{0}^{5}[f(x)+g(x)] d x = 6 + 2 = 8\) and \(\int_{0}^{5}[f(x)-g(x)] d x = 6 - 2 = 4\)

Step 2: Evaluate the integral of a multiple of a function

To evaluate the integral of a multiple of a function like \(\int_{0}^{5}-4 f(x) d x\), use the property \( \int a*f(x) dx = a*\int f(x) dx\). Given that \(\int_{0}^{5} f(x) d x=6\), the definite integral becomes \(\int_{0}^{5}-4 f(x) d x = -4 * 6 = -24\)

Step 3: Evaluate the integral of a linear combination of functions

To evaluate the integral of a linear combination of functions like \(\int_{0}^{5}[f(x)-3 g(x)] d x\), use the property \(\int [a*f(x) \pm b*g(x)] dx = a*\int f(x) dx \pm b*\int g(x) dx\). Given that \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\), the definite integral becomes \(\int_{0}^{5}[f(x)-3 g(x)] d x = 6 - 3*2 = 0\)