How do I find the arc length of the curve #y=ln(cos(x))# over the interval #[0,π/4]#?

Answer 1
The answer is: #ln(sqrt2+1)#
To find the lenght of a curve #L#, written in cartesian coordinates, it is necessary to use this formula:
#L=int_a^bsqrt((1+[f'(x)]^2))dx#.
Since #f'(x)=1/cosx*(-sinx)#, then:
#L=int_0^(pi/4)sqrt(1+(sin^2x)/(cos^2x))dx=int_0^(pi/4)sqrt((cos^2x+sin^2x)/(cos^2x))dx=int_0^(pi/4)sqrt(1/(cos^2x))dx=int_0^(pi/4)1/cosxdx#.

This integral has to be done using this substitution (parametric formulae):

#t=tan(x/2)rArrx/2=arctantrArrx=2arctanxrArrdx=2/(1+t^2)dt#,
and it's known that: #cosx=(1-t^2)/(1+t^2)#,
if #x=0# then #t=0#
if #x=pi/4# then #t=tan(pi/8)=sqrt2-1#

So:

#int_0^(sqrt2-1)1/cosxdx=int_0^(sqrt2-1)1/((1-t^2)/(1+t^2))2/(1+t^2)dt=2int_0^(sqrt2-1)1/(1-t^2)dt#,
#1/(1-t^2)=1/((1+t)(1-t))=A/(1+t)+B/(1-t)=(A(1-t)+B(1+t))/((1+t)(1-t))#,
Two polynomials (#1# on the left and #[A(1-t)+B(1+t)]# on the right) are identical if they assume the same values at the same values of #t#:
If #t=-1# then #1=A*2rArrA=1/2#;
If #t=1# then #1=B*2rArrB=1/2#.

The integral becomes:

#2int_0^(sqrt2-1)[(1/2)/(1+t)+(1/2)/(1-t)]dt=2(1/2)int_0^(sqrt2-1)[1/(1+t)-(-1)/(1-t)]dt=[ln|1+t|-ln|1-t|]_0^(sqrt2-1)=ln|1+sqrt2-1|-ln|1-sqrt2+1|=lnsqrt2-ln(2-sqrt2)=ln(sqrt2/(2-sqrt2))=ln(sqrt2/(2-sqrt2)*(2+sqrt2)/(2+sqrt2))=ln((2sqrt2+2)/(4-2))=ln((2(sqrt2+1))/2)=ln(sqrt2+1)#.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the arc length of the curve y = ln(cos(x)) over the interval [0, π/4], you can use the arc length formula:

Arc Length = ∫[a,b] √(1 + (dy/dx)^2) dx

In this case, the interval is [0, π/4], and the function is y = ln(cos(x)). You'll need to find the derivative dy/dx and then plug into the formula to calculate the arc length.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7