Combining linearity and additivity

Suppose $f$ and $g$ are functions defined on $[0,6]$ satisfying $\int_0^6 f(x)\,dx = 10, \qquad \int_0^4 f(x)\,dx = 3, \qquad \int_0^6 g(x)\,dx = -2, \qquad \int_0^4 g(x)\,dx = 5.$

Using the given information, evaluate $\int_4^6 \left[2f(x) - 3g(x)\right] dx.$

Hint

First use additivity over adjacent intervals to find $\int_4^6 f(x)\,dx$ from the two integrals of $f$ . Similarly, find $\int_4^6 g(x)\,dx$ from the two integrals of $g$ . Then apply linearity to split the combined integral into separate integrals of $f$ and $g$ over $[4,6]$ .

Step-by-step solution

Since $\int_0^6 f(x)\,dx = \int_0^4 f(x)\,dx + \int_4^6 f(x)\,dx$ , we get $10 = 3 + \int_4^6 f(x)\,dx$ , so $\int_4^6 f(x)\,dx = 7$ .

Similarly, since $\int_0^6 g(x)\,dx = \int_0^4 g(x)\,dx + \int_4^6 g(x)\,dx$ , we get $-2 = 5 + \int_4^6 g(x)\,dx$ , so $\int_4^6 g(x)\,dx = -7$ .

By linearity, $\int_4^6 [2f(x) - 3g(x)]\,dx = 2\int_4^6 f(x)\,dx - 3\int_4^6 g(x)\,dx = 2(7) - 3(-7) = 14 + 21 = 35$ .

Combining linearity and additivity

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