Question

let f be a continuous function such that \\(\int_{8}^{3} f(x) dx = -2\\) and \\(\int_{8}^{10} f(x) dx = 10\\). what is the value of \\(\int_{3}^{10} f(x) dx\\)?

Explanation:

Step1: Recall the property of definite integrals

The property of definite integrals states that \(\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx = \int_{a}^{c}f(x)dx\). Rearranging this formula, we get \(\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx-\int_{a}^{b}f(x)dx\). In our case, we want to find \(\int_{3}^{10}f(x)dx\), and we know \(\int_{8}^{3}f(x)dx=- 2\) (note that \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\), so \(\int_{8}^{3}f(x)dx =-\int_{3}^{8}f(x)dx=-2\) implies \(\int_{3}^{8}f(x)dx = 2\)) and \(\int_{8}^{10}f(x)dx = 10\). Also, using the property \(\int_{3}^{10}f(x)dx=\int_{3}^{8}f(x)dx+\int_{8}^{10}f(x)dx\). Alternatively, we can use the property \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\), so if we let \(a = 3\), \(b=8\), \(c = 10\), then \(\int_{3}^{10}f(x)dx=\int_{3}^{8}f(x)dx+\int_{8}^{10}f(x)dx\). We know that \(\int_{8}^{3}f(x)dx=-2\), so \(\int_{3}^{8}f(x)dx=-\int_{8}^{3}f(x)dx = 2\) (because \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)).

Step2: Substitute the known values

We know that \(\int_{3}^{8}f(x)dx = 2\) and \(\int_{8}^{10}f(x)dx=10\). Then, by the property of definite integrals \(\int_{3}^{10}f(x)dx=\int_{3}^{8}f(x)dx+\int_{8}^{10}f(x)dx\). Substituting the values, we get \(\int_{3}^{10}f(x)dx=2 + 10\).

Step3: Calculate the result

\(2+10 = 12\).

Answer:

\(12\)