Find the area under the curve y = (4x^3) + (9x^2) - 2x + 7 between x=0 and x=2

To be able to solve this, the equation needs to be integrated. To do this, simple integration needs to be applied where the power of 'x' increases by 1 and the coefficient of 'x' is divided by the new power. The equation below is equal to that given in the question y = 4(x^3) + 9(x^2) - 2(x^1) + 7(x^0). This makes it more clear how the integration is carried out to give x^4 + 3(x^3) - x^2 + 7x. To find the area, the new equation needs to be solved by substituting in the limits, therefore x=2 and x=0. For x=2, ((2)^4) + 3((2)^3) - (2)^2 + 7(2) = 16 + 24 - 4 + 14 = 50 For x=0, ((0)^4) + 3((0)^3) - (0)^2 + 7(0) = 0. To find the area, the value for the equation in which the lower limit of x was substituted(therefore 0) needs to be subtracted from the one with the higher limit(therefore 2). Hence the area is given by: area = 50 - 0 = 50 square units.