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

Answer:

Arc length=$\int_{0}^{\pi}\sqrt{1+20.25sin^2(4.5x)}dx$

Step-by-step 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 $\frac{d(cos ax}{dx}=-asin(ax)$

Arc length=$\int_{a}^{b}\sqrt{1+f'(x)^2}}dx$

Substitute the values

Arc length=$\int_{0}^{\pi}\sqrt{1+(4.5sin 4.5x)^2}dx$

Arc length=$\int_{0}^{\pi}\sqrt{1+20.25sin^2(4.5x)}dx$