7. if $\\int_{4}^{-10} g(x) d x=-3$ and $\\int_{4}^{6} g(x) d x=5$, then $\\int_{-10}^{6} g(x) d x=$\n(a)…

7. if $\\int_{4}^{-10} g(x) d x=-3$ and $\\int_{4}^{6} g(x) d x=5$, then $\\int_{-10}^{6} g(x) d x=$\n(a) $-8$\n(b) $-2$\n(c) $2$\n(d) $8$

7. if $\\int_{4}^{-10} g(x) d x=-3$ and $\\int_{4}^{6} g(x) d x=5$, then $\\int_{-10}^{6} g(x) d x=$\n(a) $-8$\n(b) $-2$\n(c) $2$\n(d) $8$

Answer

Explanation:

Step1: Use the property of definite integrals

We know that (\int_{a}^{c}g(x)dx=\int_{a}^{b}g(x)dx+\int_{b}^{c}g(x)dx). Let (a = - 10), (b = 4), (c = 6). Then (\int_{-10}^{6}g(x)dx=\int_{-10}^{4}g(x)dx+\int_{4}^{6}g(x)dx). Also, (\int_{a}^{b}g(x)dx=-\int_{b}^{a}g(x)dx), so (\int_{-10}^{4}g(x)dx=-\int_{4}^{-10}g(x)dx). Since (\int_{4}^{-10}g(x)dx=-3), then (\int_{-10}^{4}g(x)dx = 3).

Step2: Calculate (\int_{-10}^{6}g(x)dx)

We are given (\int_{4}^{6}g(x)dx = 5). Substitute (\int_{-10}^{4}g(x)dx = 3) into (\int_{-10}^{6}g(x)dx=\int_{-10}^{4}g(x)dx+\int_{4}^{6}g(x)dx). We get (\int_{-10}^{6}g(x)dx=3 + 5).

Answer:

(8), so the answer is (D).