Question

The functions f and g are integrable and , , and . Evaluate the integral below or state that there is not enough information.

Answer 1

The integral
`\(\int_a^b (f(x) - g(x)) \, dx\)` evaluates to 4.

Step-by-step explanation:

Given
`\(f(x) \geq g(x) \geq 0\)` over [a, b] and
`\(\int_a^b f(x) \, dx = 10\)` and
`\(\int_a^b g(x) \, dx = 6\)`, it's evident that
`\(f(x) - g(x) \geq 0\)` over the interval. Therefore,
`\(\int_a^b (f(x) - g(x)) \, dx\)` represents the difference between the integrals of f and g over [a, b]. Considering that
`\(f(x) - g(x) \geq 0\)`, this difference in integrals is 10 - 6 = 4.

The integral of the difference between two functions f(x) and g(x) over a closed interval [a, b] represents the net accumulation of the difference between the areas above g(x) and below f(x) within that interval. Here, since
`\(f(x) \geq g(x) \geq 0\)` for all x in [a, b], the integral
`\(\int_a^b (f(x) - g(x)) \, dx\)` measures the surplus area between the curves of f and g over the given interval.

Here is complete question;

"The functions f and g are integrable over a closed interval [a, b] such that
`\(f(x) \geq g(x) \geq 0\)` for all x in [a, b], and
`\(\int_a^b f(x) \, dx = 10\)` and
`\(\int_a^b g(x) \, dx = 6\)`. Evaluate the integral below or state that there is not enough information:

`\[\int_a^b (f(x) - g(x)) \, dx\]`"

Answer 2

"The integral of the sum of functions f(x) and g(x) from (a) to (b) is equal to the sum of the integrals of f(x) and g(x) over the same interval."

Step-by-step explanation:

The given integral is the sum of two integrals:
`\(\int_(a)^(b) f(x) \,dx\)` and
`\(\int_(a)^(b) g(x) \,dx\)`. By the linearity property of integration, we can separate the integral of the sum into the sum of integrals. This property states that
`\(\int_(a)^(b) [f(x) + g(x)] \,dx = \int_(a)^(b) f(x) \,dx + \int_(a)^(b) g(x) \,dx\)`. Therefore, the final answer is the sum of the individual integrals of f(x) and g(x).

In the first integral,
`\(\int_(a)^(b) f(x) \,dx\)`, we evaluate the antiderivative of f(x) with respect to (x) at the upper limit (b) and subtract the value at the lower limit (a). Similarly, in the second integral,
`\(\int_(a)^(b) g(x) \,dx\)`, we find the antiderivative of g(x) and apply the limits of integration. The final result is the sum of these two antiderivative evaluations, providing the value of the given integral.

In conclusion, the integral
`\(\int_(a)^(b) (f(x) + g(x)) \,dx\)` can be computed by separately evaluating the integrals of f(x) and g(x) over the given limits and summing the results. This approach is based on the linearity property of integration, allowing us to break down the integral of a sum into the sum of integrals.