Evaluate the line integral \int_C (8x+9z) ds, where C is the curve x=t, y=2t^2, z=t^3, 0 \leq...

Question:

Evaluate the line integral {eq}\displaystyle\int_C (8x+9z) \: ds {/eq}, where {eq}C {/eq} is the curve {eq}x=t, \ \ y=2t^2, \ \ z=t^3, \ \ 0 \leq t \leq 1 {/eq}. Show all steps.

Evaluating a Line Integral:

The line integral {eq}\displaystyle\int_C f(x, y) \: ds {/eq} along the curve {eq}C {/eq} defined parametrically: {eq}\mathbf{r}(t) = f(t) \: \mathbf{i} + g(t) \: \mathbf{j} + h(t) \: \mathbf{k}, \: a \leq t \leq b {/eq} is defined as {eq}\displaystyle\int_C f(x, y) \: ds = \int_a^b f(x(t), y(t)) \: \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} \: dt. {/eq} This can be challenging to determine, depending on the radicand inside the integrand.

Answer and Explanation:

We are given {eq}\displaystyle\int_C (8x+9z) \: ds {/eq}, where {eq}C {/eq} is the curve {eq}x=t, \ \ y=2t^2, \ \ z=t^3, \ \ 0 \leq t \leq...

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Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2
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