Question

Find the length of the function over the given interval. x = 2y from y = −1 to y = 1

Answer 1

The length of the function x = 2y is 2√5 units

According to the given question.

We have a function

x = 2y

Since, we know that the length of the function in the interval [a, b] is given by

l =    $\int\limits^a_b\sqrt{1+(\frac{dx}{dy}) ^{2} } \,dy$

Where, l is the length of the function.

Now, differentiating the given function x = 2y w.r.t y.

Therefore,

dx/dy = 2

So, the length of the given function x = 2y  from y = -1 to y = 1 is given by

l =   $\int\limits^1_{-1}{\sqrt{1+2^{2} } } \,dy$

⇒ l =   $\int\limits^1_{-1} {\sqrt{1+4} } \,dy$

⇒ l =   $\int\limits^1_{-1}{\sqrt{5} } \,dy$

⇒ l = √5[1 - (-1)]

⇒ l = √5(2)

⇒ l = 2√5

Hence, the length of the function x = 2y is 2√5 units.

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