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question 16
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the values of several definite integrals are given as follows:
\\(\int_{0}^{2} f(x) dx = 5\\) \\(\int_{0}^{3} f(x) dx = 7\\) \\(\int_{0}^{3} g(x) dx = -3\\) \\(\int_{3}^{3} g(x) dx = 5\\)
use these values and properties of definite integrals to evaluate the indicated definite integral.
find a formula for \\(a\\) in terms of \\(b\\) such that \\(\int_{0}^{3} (af(x) + bg(x)) dx = 0\\).
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Explanation:

Step1: Use integral properties

Use the property \(\int_{0}^{3}(af(x)+bg(x))dx=a\int_{0}^{3}f(x)dx + b\int_{0}^{3}g(x)dx\). Also, note that \(\int_{0}^{3}g(x)dx=\int_{0}^{2}g(x)dx+\int_{2}^{3}g(x)dx\). Given \(\int_{0}^{2}g(x)dx=- 3\) and \(\int_{2}^{3}g(x)dx = 5\), then \(\int_{0}^{3}g(x)dx=-3 + 5=2\). And \(\int_{0}^{3}f(x)dx = 7\).

Step2: Substitute into the integral equation

Substitute into \(a\int_{0}^{3}f(x)dx + b\int_{0}^{3}g(x)dx=0\). We get \(a\times7+b\times2 = 0\).

Step3: Solve for \(a\)

Solve the linear equation \(7a+2b = 0\) for \(a\). Subtract \(2b\) from both sides: \(7a=-2b\). Then divide both sides by \(7\): \(a=-\frac{2}{7}b\).

Answer:

\(a =-\frac{2}{7}b\)