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we're given that the definite integral from one to four of f of X DX is equal to 6 and the definite integral from 1 to 7 of f of X DX is equal to 11 and we want to figure out the definite integral from 4 to 7 of f of X DX so at least in my brain I'm visualizing these as areas between the curve y equals f of X and the x-axis and so let's just draw that we don't know exactly what f of X is but we can draw an arbitrary f of X just to help us visualize things so let me draw so if that is drive it in a bolder color so that's our y-axis and this is our x-axis and let's see all the action is happening between X from X is going from 1 to 7 so we could go 1 I mean we can go 1 2 3 4 5 6 7 and we can even go to 8 if we like but the important numbers see we're dealing with 1 2 3 4 5 6 7 and then we go to 8 and let me just draw the graph y equals f of X and I'm just going to draw something arbitrary here so let's say the graph of y equals f of X looks like that y is equal to f of X of course I let me label my axes that's the x axis that is the y axis now let's think about what each of these integrals represent so the integral the definite integral from 1 to 4 well that's going to be we're going to be going from 1 to 4 right over here so this is the definite integral from 1 to 4 this area in under the curve between the curve and the x-axis DX which is equal to 6 and now let's see we also have the region that goes from from 4 to 7 we have this region right over here and that that area is represented by this definite integral the one that we need to figure out the definite integral from 4 to 7 of f of X DX we need to figure that out and they also what else do we have so let me underline this so that's that this is the area of the region between x equals 4 and x equals 7 under y equals f of X above the x-axis and then they also gave us this last piece of information which is let me do this in another color the definite integral from 1 to 7 well that's going from 1 1 all the way to 7 so that's the sum of these two regions right over here so we could rewrite this as the definite integral from 1 to 4 of f of X DX plus plus the definite integral from 4 to 7 of f of X DX is equal to is equal to the definite integral from 1 to 7 1 to 7 of f of X DX and notice what's going on here this the first one just goes to the area from 1 to 4 then we go from 4 to 7 so if you add those together that's going to be the area from 1 to 7 from 1 to 7 and so they give us a lot of this information they tell us that this right over here is 6 let me do that same color they tell us that this is 6 they tell us that this is 11 and so we have 6 plus this is equal to 11 well 6 plus what is equal to 11 well this thing right over here must be equal to this thing right the the definite integral from 4 to 7 must be equal to 5 this must be equal to 5 another way to think about it if this region right over I'm having trouble switching colors if this region right over here is 6 so that has an area of 6 and the whole region if everything has an area of 11 so if that Plus that has an area of 11 well then the stuff that we don't know this orange this orange region this orange region is going to be 11 - six so this region right over here is going to have an area of five