The answer is I≈57.7 is it true? Help me check the answer. thank you

Question

If C is given by x=2cost, y=2sint, z=πt, 0≤t≤π, evaluate : ∫_C(x+z)ds.

Answer 1

Given:

x(t) = 2 cos( t )

y(t) = 2 sin( t )

z(t) = πt

∀ t ∈ [ 0, π ] (fancy math lingo for "For all" t "that is within" "the closed set between 0 to π")

Find:

I = ∫_C ( x + z ) ds = ?

Solution:

ds is the small increment of distance along the path defined by C and parameterized by t

In this case, ds is 3-dimensional.

If you describe some position vector s from the origin like, s = x i + y j + z k, you'll get:

s( t ) = 2 cos( t ) i + 2 sin( t ) j + πt k

Inside of multi-variable calculus, you can take the differential operator 'd' on that vector and see what the tangent vector ds would look like defined by the path C and parameter t

Note: the differential operator 'd' works like normal derivatives of functions but you "abuse" notation by doing the following:

ds/dt * dt = d/dt ( s ) * dt

ds = d/dt ( s ) * dt

ds = ( -2 sin( t ) i + 2 cos( t ) j + π k ) dt

We arrive at an issue though because of the fact that the integral defined above doesn't deal with a vector ds, it deals with its magnitude, ds

(math lingo for 'therefore' ) ∴ ds = √( ( -2 sin( t ) )² + ( 2 cos( t ) )² + ( π )² ) dt

ds = √( 4 * [ ( sin( t ) )² + ( cos( t ) )² ] + ( π )² ) dt

From a helpful trig identity: ( sin( t ) )² + ( cos( t ) )² = 1

ds = √( 4 + π² ) dt

The bounds of the integral are the bounds of the closed set for t

I = ∫_C ( x + z ) ds = ∫₀^π ( 2 cos( t ) + πt ) √( 4 + π² ) dt

√( 4 + π² ) has no dependency on t so it can come out of the integral and scale whatever answer we get from evaluating a "simpler" integral

I = √( 4 + π² ) * ∫₀^π ( 2 cos( t ) + πt ) dt

I'm having trouble recalling which property it is about integrals but the following is true:

The integral of a sum is equal to the sum of integrals

I = √( 4 + π² ) * [ 2 * ∫₀^π ( cos( t ) ) dt + π * ∫₀^π ( t ) dt ]

I = √( 4 + π² ) * [ 2 * sin( t )|₀^π + π/2 * t²|₀^π ]

I = √( 4 + π² ) * [ 2 * ( 0 - 0 ) + π/2 * ( π² - 0² ) ]

I = √( 4 + π² ) * [ π³/2 ]

I = π³/2 * √( 4 + π² ) ≈ 57.7

I hope this helps! Please message me in the comments if you have any questions, comments, or concerns!