Question

Write the integral that gives the length of the curve y equals f (x )equals Integral from 0 to 4.5 x sine t dt on the interval ​[0,pi ​].

Answer 1

Arc length=
`\int_(0)^(\pi)√(1+20.25sin^2(4.5x))dx`

Explanation:

We are given that

`y=f(x)=\int_(0)^(4.5x) sin t dt`

We know that
`\int sin x=-cos x+C`

Using the formula

`f(x)=[-cos x]^(4.5x)_(0)`

`f(x)=-cos (4.5x)+cos 0=-cos 4.5 x+1`

Because cos 0=1

f'(x)=4.5 sin 4.5 x

Because
`(d(cos ax)/(dx)=-asin(ax)`

Arc length=
`\int_(a)^(b)√(1+f'(x)^2)}dx`

Substitute the values

Arc length=
`\int_(0)^(\pi)√(1+(4.5sin 4.5x)^2)dx`

Arc length=
`\int_(0)^(\pi)√(1+20.25sin^2(4.5x))dx`