# Line integral, problems with substitution (probably!)

#### Saraphim

1. Homework Statement

I am trying to solve a line integral (bear with me, I am new to calculus!) and my basic skills of integration seem to fail me. I am sure the mistake is quite obvious, as I keep getting the wrong answer, 2, when it should be ~2.69

2. Homework Equations
$$\int_C 4x^3dS$$

C is the curve given by $$x=t, y=t^3-1$$ and $$0 \leq t \leq 1$$

3. The Attempt at a Solution
$$\int_C 4x^3 dS = \int_0^1 4t^3 \sqrt{1^2+(3t^2)^2} dt = \int_0^1 4t^3 \sqrt{9t^4+1} dt$$
I attempt substitution in order to solve the integral:
$$u=9t^4+1 \Rightarrow \frac{1}{36}du=t^3 dt$$.
The limits are now $$9 \cdot 0^4=0$$ and $$9 \cdot 1^4=9$$, so by substitution we have:
$$\frac{4}{36} \int_0^9 \sqrt{u} \, du = \frac{1}{9} \left[\frac{2}{3} u^{\frac{3}{2}}\right]_0^9 = \frac{2}{27} \left(9^{\frac{3}{2}} - 0\right)$$

Which equals.... Two! Well, there's a mistake in there somewhere, so, no it doesn't. Hopefully a non-mathematician will have mercy on me so that the punishment isn't too hard.

-- Sarah

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#### Saraphim

Nevermind, I found the problem. I did not correctly calculate the new limits, they should be 10 and 1 respectively.

-- Sarah

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