Answer:
To calculate the value of the definite integrals requested, we can use the properties of integrals and the given information about the integrals of the functions \(f(x)\) and \(g(x)\). We will break down each integral step by step.
a) \( \int_{-5}^{3} f(x) dx \)
We can find this integral by subtracting the integral of \(f(x)\) from -5 to 8 from the integral of \(f(x)\) from -5 to 8. Using the given information:
\[
\int_{-5}^{3} f(x) dx = \int_{-5}^{8} f(x) dx - \int_{3}^{8} f(x) dx
\]
Now, we can use the given values to calculate this:
\[
\int_{-5}^{3} f(x) dx = 16 - (-24) = 16 + 24 = 40
\]
So, \( \int_{-5}^{3} f(x) dx = 40 \).
b) \( \int_{-5}^{8} (2f(x) - g(x)) dx \)
In this case, we are asked to find the integral of \(2f(x) - g(x)\) from -5 to 8. We can distribute the integral across the sum:
\[
\int_{-5}^{8} (2f(x) - g(x)) dx = 2\int_{-5}^{8} f(x) dx - \int_{-5}^{8} g(x) dx
\]
Now, we can use the given values to calculate this:
\[
\int_{-5}^{8} (2f(x) - g(x)) dx = 2(16) - 3.5 = 32 - 3.5 = 28.5
\]
So, \( \int_{-5}^{8} (2f(x) - g(x)) dx = 28.5 \).
c) \( \int_{-5}^{3} (f(x) + g(x)) dx \)
Here, we need to find the integral of \(f(x) + g(x)\) from -5 to 3. We can again distribute the integral:
\[
\int_{-5}^{3} (f(x) + g(x)) dx = \int_{-5}^{3} f(x) dx + \int_{-5}^{3} g(x) dx
\]
Now, using the given values:
\[
\int_{-5}^{3} (f(x) + g(x)) dx = 40 + \left(\int_{-5}^{8} g(x) dx - \int_{3}^{8} g(x) dx\right) = 40 + (3.5 - (-24)) = 40 + 27.5 = 67.5
\]
So, \( \int_{-5}^{3} (f(x) + g(x)) dx = 67.5 \).
Explanation: