Tim W. answered • 08/21/21
Experienced and enthusiastic university instructor and math tutor
Unfortunately this problem cannot be solved using only the given information. Was there additional information given? Or is there possibly a typo in the problem statement? Is the second integral supposed to have limits of -4 to -1 rather than from 0 to 3?
I'll write the solution to that problem in case that is helpful. In other words, I'll give the solution to the following question:
"Given that ∫03 f(x)dx = 8, find ∫-4-1 f(x+4)dx."
This problem can be solved with integration by substitution (u-substitution). Using the substitution u=x+4 on the integral ∫-4-1 f(x+4)dx, we get du/dx=1, so du=dx. We must then transform the x-limits of integration (of -4 and -1) to u-limits. Since u=x+4, we see that if x=-4, then u=0. Similarly, the upper limit of x=-1 becomes u=-1+4=3.
Making the substitutions into the integral, we obtain: ∫-4-1 f(x+4)dx= ∫03 f(u)du. This integral is the same as the given value of ∫03 f(x)dx = 8 (just using the letter u instead of x). So we have ∫-4-1 f(x+4)dx= ∫03 f(u)du=8.
Once again, I'm not sure this is what you're after since your original question is unsolvable as written. But hopefully working through this alternate example will help with the concepts you're learning!
