This is a question that is bugging me because trying to solve it the spirit of the topic I am on (backed up by one example under simple circumstances and a frustratingly complicated explaination) is proving really difficult. Trying to tackle it on my own terms yields me a wrong answer A curve is given parametrically: x= t^3, y= t^2 a tangent to the curve is drawn at the point where t=2 find a) the equation of the tangent b) the area bounded between the curve, tangent and y-axis (a) isn't a problem..the answer is 3y = x+4 (b) is a problem though. My first inclination is to express y in terms of x, if x = t^3 then x^(2/3) = t^2 and so y=x^(2/3) or 3y =3(x^(2/3)) now I have two equations with the same variables that I should be able to integrate without a problem subtracting 3(x^(2/3)) from x+4 I get: x+4 - 3(x^(2/3))... integrating this w.r.t.x I get [1/2(x^2) +4x - 9/5(x^(5/3))] as x = t^3, if t = 2 then x = 8 and so my limits should be 0 and 8 plugging x=8 into the above however yields an answer of 6.4, the books answer is 2.13 (and by plotting the damned thing and measuring trapeziums I find that the book answer is correct) solving it in terms of t though is mega uncomfortable...the book example isn't very helpful and I just don't know what I'm doing. am I right that the tangent can be expressed as 3t^2=t^3 +4? Even if this is true how do I work with the curve?