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## Homework Statement

My question is really regarding a shortcut. I never did displacement/distance and velocity with integrals in Calculus 1 and it's time for review. The professor offers a way to find the two new definite integrals in

**distance traveled**by creating an elaborate diagram with a number line and lots of pictures and boxes, but I notice with each question I do that the relevant number to put into the definite integral b and a is just the root that between the number line of the original integral.

## Homework Equations

Example is integral at b=6 and a=1, equation t^2-2t-8. The roots of this are -2 and 4. Since 4 is in between 6 and 1, I picked 4, so my two new definite integrals are b=4 a=1 and b=6 a=4 in the first and second equations and I solve the definite integral accordingly for 98/3m^3

My question is this: Are these diagrams and number lines she draws out really necessary to find anything or can I always just find b and a of the two new definite integrals by finding which root of the equation is in between the numbers of the original definite integral.

I apologize for not using the math notations; I can't seem to get them to work for me the way I want to. I am also sorry if my question appears vague.