Question

Find the value of integral ∫C(x2+y2+z2)ds∫C(x2+y2+z2)ds, where CC is parmeterized by →r(t)=〈3cos(t),3sin(t),2t〉r→(t)=〈3cos(t),3sin(t),2t〉 for 0≤t≤20≤t≤2. Round your answer to four decimal places.

Answer 1

To find the value of the given integral, substitute the parameterization of the curve C into the integrand and integrate over the given interval. The approximate value of the integral is 25.3333.

Step-by-step explanation:

To find the value of the given integral, we need to parameterize the curve C and substitute it into the integrand. The curve C is parameterized by:

r(t) = (3cos(t), 3sin(t), 2t)

Substituting this into the integrand, we get:

(x^2 + y^2 + z^2)ds = (9cos^2(t) + 9sin^2(t) + 4t^2)dt

Now, we can integrate this expression over the interval 0 ≤ t ≤ 2:

∫(9cos^2(t) + 9sin^2(t) + 4t^2)dt = ∫(9 + 4t^2)dt = 9t + (4/3)t^3

Substituting the limits of integration, we have:

∫C(x^2 + y^2 + z^2)ds ≈ 9(2) + (4/3)(2^3) ≈ 25.3333