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(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Definite integration problem

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