a line L intersects (0, 0) and the curve y=x^2 at x=t. What is the equation of the line? What is the area between L and y from x=0 to x=t?


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Question Number 8070 by FilupSmith last updated on 29/Sep/16

$a line L intersec t s (0, 0) a n d t h e c u r v e$ $y = x^{2} a t x = t .$ $What is the equation of the line ?$ $What is the area between L and y from$ $x = 0 to x = t ?$
	Answered by sandy_suhendra last updated on 29/Sep/16

	$s o t h e l i n e L i n t e r s e c t s (0, 0) a n d (t, t^{2})$ $t h e e q u a t i o n o f L \Rightarrow \frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}}$ $\frac{x - t}{0 - t} = \frac{y - t^{2}}{0 - t^{2}}$ $t x - t^{2} = y - t^{2}$ $y = t x$ $t h e a r e a b e t w e e n L a n d y$ $=_{0} \int^{t} (t x - x^{2}) d x$ $= {[\frac{1}{2_{}} {t x}^{2} - \frac{1}{3} x^{3}]}_{0}^{t}$ $= \frac{1}{2} t^{3} - \frac{1}{3} t^{3} = \frac{1}{6} t^{3}$
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