Final answer:
The question asks for the bounds of the integral of f(x) over the interval [-1,3], given that f(x) is between -2 and 5. The upper and lower bounds are calculated by considering the maximum and minimum values f(x) could take. However, the calculated bounds do not match any of the provided options, suggesting a need for reevaluation.
Step-by-step explanation:
The question deals with finding the bounds for an integral given a range for the function f(x). The function f(x) is bounded by −2 and 5 over the interval [−1,3], which means that at any point in this interval, the value of f(x) will not exceed 5 or be less than −2. The integral from −1 to 3 of f(x) dx represents the net area under the curve of f(x) between x = −1 and x = 3.
To find the bounds, we consider the maximum and minimum values that f(x) can take. If f(x) were to take its maximum value of 5 over the whole interval, the integral would be the area of a rectangle with a height of 5 and a width of 4 (since 3 − (−1) = 4), giving us an upper bound of 20. Similarly, if f(x) took its minimum value of −2 over the whole interval, the integral would give an area (technically, a negative area since it's below the x-axis) with a height of −2 and a width of 4, giving us a lower bound of −8. However, these bounds are not listed in the options provided. We must, therefore, adjust for the true width by looking at the actual difference from the upper and lower bounds of f(x), which are 5 and −2, respectively. This gives an upper bound of (5 − (−2)) × 4 = 7 × 4 = 28 and a lower bound of 0, since the smallest value is non-negative. So, the correct bounds for the integral are
0 ≤ ∫−13f(x)dx ≤ 28
However, these specific bounds are not listed in the options provided, so we should take another look at the question and the options given to ensure we didn't make an error in our calculation or interpretation.