f(x)= 3sqrt(x-9) ---------- the three is small right above the sqrt
g(x)= (x − 9)
thnxs so much in advance
g(x)= (x − 9)
thnxs so much in advance
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ok so for f(x) i'm assuming you mean that it's a CUBE root then.
so first off, graph both the equations on a graphing calculator, but if you can't use one, just you can still follow these steps by hand for the most part:
1. graph on calculator or by hand
2. determine which function is on "top" of the other.
3. in this case, the two graphs "take turns" being on top, so you have to take the integral of two separate pieces and then add them to get the total area.
4. start by taking the integral of the bottom piece. so the g(x) (the line) is on top of the f(x) curve first. so you would solve for the integral of g(x)-f(x), or in other words (x-9)-[(x-9)^1/3]. the 1/3 is another way to write the cube root. i solved the integral on the calculator, but you can do the same by doing it by hand.
5. you also need the limits of the area enclosed, so you need to find the intersection points. again i will calculate them with the calculator. and they are: (8, -1) and at the x-axis at (9,0). the x values are the ones you need, so it would be 8 and 9.
6. write out the equation you need to solve for this first piece. it would be the integral from 8 to 9 of g(x)-f(x) which is also (x-9)-[(x-9)^1/3].
7. so you can calculate that by hand or calculator. i got .25003796.
8. you can estimate that the second piece above the x-axis is also the same area, but just do it again to double check, but this time, the f(x) graph is on top. and you also have different limits, 9 and 10.
9. so do the integral from 9 to 10 of f(x)-g(x), or [(x-9)^1/3]-(x-9)
10. it came out the same for this area, .25003796.
so in the end the total area would be 2 times .25003796...which is .5000759, close to just 1/2.
so first off, graph both the equations on a graphing calculator, but if you can't use one, just you can still follow these steps by hand for the most part:
1. graph on calculator or by hand
2. determine which function is on "top" of the other.
3. in this case, the two graphs "take turns" being on top, so you have to take the integral of two separate pieces and then add them to get the total area.
4. start by taking the integral of the bottom piece. so the g(x) (the line) is on top of the f(x) curve first. so you would solve for the integral of g(x)-f(x), or in other words (x-9)-[(x-9)^1/3]. the 1/3 is another way to write the cube root. i solved the integral on the calculator, but you can do the same by doing it by hand.
5. you also need the limits of the area enclosed, so you need to find the intersection points. again i will calculate them with the calculator. and they are: (8, -1) and at the x-axis at (9,0). the x values are the ones you need, so it would be 8 and 9.
6. write out the equation you need to solve for this first piece. it would be the integral from 8 to 9 of g(x)-f(x) which is also (x-9)-[(x-9)^1/3].
7. so you can calculate that by hand or calculator. i got .25003796.
8. you can estimate that the second piece above the x-axis is also the same area, but just do it again to double check, but this time, the f(x) graph is on top. and you also have different limits, 9 and 10.
9. so do the integral from 9 to 10 of f(x)-g(x), or [(x-9)^1/3]-(x-9)
10. it came out the same for this area, .25003796.
so in the end the total area would be 2 times .25003796...which is .5000759, close to just 1/2.