00:01
We are given a curve and we're asked to find the exact length of this curve.
00:06
The curve is given by the equation y squared is equal to 4 times x plus 4 cubed.
00:19
We have the x lies in the interval 0, 2, and that y is greater than 0.
00:39
So, because y is greater than 0, it follows that this equation can also be written as y is equal to well, the square root of both sides, 2 times x plus 4 to the 3 halves.
00:58
And again, we have that x lies between 0 and 2.
01:07
And therefore, it follows that the derivative, d .y, d .x, this is going to be 2 times 3 halves is 3 times x plus 4 to the 3 halves minus 1 is 1 half.
01:31
And therefore we have that 1 plus d, y, dx squared is 1 plus 9 times x plus 4, which, multiplying out, we get this is the same as 9x plus 37.
01:59
Therefore, the length of this curve is the integral from 0 to 2 of the square root of 9...