Question

How do you do this question?

Answer 1

(8√2) / 15

Explanation:

A curve bounded by the y-axis is represented by in terms of dy;

`\int \:x\:dt`

When the curve crosses the y-axis, x will be 0. In this case x is the function of t, so we have to solve for x(t) = 0;

0 = t^2 + 2t --- (1)

Solution(s) => t = 0, t = 2

dy = (1/2 * 1/√t)dt --- (2)

Our solutions (0, 2) are our limits. The area of the curve is in the form
`A\:=\:\int _b^a\:f\left(t\right)g'\left(t\right)dt` , so now let's introduce the limits of integration, x(t) and dy/dt. Remember, dy/dt = (1/2 * 1/√t) (second equation). 1/2 * 1/√t can be rewritten as 1/2 * t^(-1/2)....

`A\:=\:\int _2^0\:\left(t^2-2t\right)\left((1)/(2)t^{-(1)/(2)}\right)dt\\\\= \int _2^0\:\left((1)/(2)t^{(3)/(2)}-t^{(1)/(2)}\right)dt\\\\= \left[\frac{t^{(5)/(2)}}{5}-\frac{2t^{(3)/(2)}}{3}\right]_2^0\\\\= 0\:-\:\left((4√(2))/(5)-(4√(2))/(3)\right)\\\\= (8√(2))/(15)`

Your solution is 8√2 / 15