Find the area under the graph between x=0 and x=2 when f(x)=x^2 +2, give your answer as an exact value.
First of all we need to find the integral of our function which is f(x). Since we are integrating with respect to x and our values are in the form of x we can use the rule of raising the power for each x term and then dividing by the new power e.g x2 would integrate to x3/3. We imaging our 2 in this case is actually 2x0 and therefore this integrates to 2x1/1 or 2x. We would then have to put a +c at the end of the function but since we are working out area it is not required and they will cancel each other out.Integral of f(x) = x3/3 +2xNow that we have our integrated function we can work out area. This is just a case of subbing in our x values and then taking away the number we get from subbing in x=2 from the number we get from subbing in x=0 (it's always the number from the larger x subbed value minus the number from the smaller x subbed value). So subbing in our x values gives us:x=2 (23/3) + (2x2) = 8/3 + 4 = 20/3x=0 (03/3) + (2x0) = 0+0 = 0Therefore the area under the graph is 20/3 - 0 = 20/3
