← Back to learning objective

Using linearity and bounds-reversal to evaluate a combined integral

Suppose ∫15f(x) dx=8\int_1^5 f(x)\,dx = 8 and ∫15g(x) dx=3\int_1^5 g(x)\,dx = 3.

Using the given information, evaluate ∫51[4f(x)−2g(x)]dx.\int_5^1 \left[4f(x) - 2g(x)\right] dx.

Hint

First use linearity to split the integral into separate integrals of ff and gg over [1,5][1,5], then apply the rule for reversing the limits of integration to switch from ∫51\int_5^1 to ∫15\int_1^5.

Step-by-step solution

By linearity, ∫51[4f(x)−2g(x)] dx=4∫51f(x) dx−2∫51g(x) dx\int_5^1 [4f(x)-2g(x)]\,dx = 4\int_5^1 f(x)\,dx - 2\int_5^1 g(x)\,dx. This splits the combined integrand into separate scalar multiples of ff and gg.

Reversing the bounds gives ∫51f(x) dx=−∫15f(x) dx=−8\int_5^1 f(x)\,dx = -\int_1^5 f(x)\,dx = -8 and ∫51g(x) dx=−∫15g(x) dx=−3\int_5^1 g(x)\,dx = -\int_1^5 g(x)\,dx = -3. Substituting, the expression becomes 4(−8)−2(−3)=−32+6=−264(-8) - 2(-3) = -32 + 6 = -26.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap