on this problem we're given the acceleration function and using these two points we want to determine velocity function and the position function how can we do this well here's some things you want to know you could find the velocity function by integrating the acceleration function and of course you need to make into account the constant C you could find a position function by integrating velocity function so that's how we could find these two functions by integration so let's start with the velocity function V of T is going to be the integral of the acceleration function well T minus six ET the antiderivative of 12T so using the power rule if you want to find the antiderivative of a variable raised to a constant it's going to be X raised to the N plus 1 divided by M plus 1 and then plus the constant C so the antiderivative of t to the first power is going to be t to the second power divided by 2. now for the constant 6 you could treat it like this so negative 6 is equivalent to negative six times t to the zero so this becomes t to the first power over one or simply just negative 16. and then we'll need to add the constant C so right now we have that V of T is going to be 6 t squared minus six t plus c now we're given a point in the velocity function V of one is equal to 10. so using this point we need to determine the constant of integration our C value so V of 1 is going to be 6 times 1 squared minus six times one plus C and we know that V of 1 is 10. so this becomes 6 minus six plus C equals 10. now 6 minus six will cancel so we get that C is equal to 10. so the first answer for this problem is this one the velocity function is six t squared minus 6t Plus 10. so now that we have that we can now calculate or determine the position function so the position function is going to be the integral of the Velocity function and so that's going to be the integral of six t squared minus six t plus ten so this is going to be 6 and then we're going to have t to the third over three the antiderivative of t to the first Power will be e squared over two and for the constant 10 when we integrate it with respect to T it's going to be 20. and then plus c so simplifying this expression we have 6 divided by three is two six divided by two is three and so forth now let's use the fact that X of 2 is equal to 17 to calculate the constant C so let's replace p with 2. we're going to set all of this equal to 17. so 2 to the third power is eight two squared is four ten times two is twenty now 2 times 8 is 16 3 times 4 is 12. 15 minus 12 is 4. 4 plus 20 is 24. so we have 24 plus C is equal to 17. subtracting both sides by 24. we get that c is equal to negative seven so we can replace C with negative seven so this is the position function so that's how we could determine the velocity function and the position function from acceleration by means of integration so when you integrate the acceleration function it will give you the velocity function and you have to use this point to solve for C once you do that you can integrate the velocity function so you get the position function once again you gotta solve for C using the second point so that's it for this video for those of you who want more example problems like this feel free to check out the links in the description section below