← 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[2f(x)−3g(x)]dx.\int_5^1 \left[2f(x) - 3g(x)\right] dx.

Hint

First use linearity to split the integral into pieces involving ∫15f(x) dx\int_1^5 f(x)\,dx and ∫15g(x) dx\int_1^5 g(x)\,dx, then handle the reversed bounds by negating the result.

Step-by-step solution

Reversing the order of the bounds introduces an overall negative sign, so ∫51[2f(x)−3g(x)]dx=−∫15[2f(x)−3g(x)]dx\int_5^1 \left[2f(x)-3g(x)\right]dx = -\int_1^5 \left[2f(x)-3g(x)\right]dx.

By linearity, ∫15[2f(x)−3g(x)]dx=2∫15f(x) dx−3∫15g(x) dx=2(8)−3(3)=7\int_1^5 \left[2f(x)-3g(x)\right]dx = 2\int_1^5 f(x)\,dx - 3\int_1^5 g(x)\,dx = 2(8) - 3(3) = 7.

Applying the negative sign from the bound reversal gives the final answer: −7-7.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap