# Thread: integrals and length of a curve

1. ## integrals and length of a curve

HI

So let curve C have equation y = 4x^(3/2)
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3
Let a > 2. The portion of C btwn the points (2, (4(sqrt 2)^2)/3) and (a,(4a^(3/2))/3) has length 49/3

Find the value of a

So if y = (4x^(3/2))/3 then dy/dx = 2x^(1/2)

Correct me anywhere you see a mistake!!

Then L = integral sign (S) a= 2 and b= a sq rt( 1 + (dy/dx)^2) dx
= S a=2 b=a sq rt (1 + 4x) dx

SO then I let u = 1 + 4x so du = 4 dx

so when x = 2 then u = 9 and when x = a then u = 1 + 4a

L = 4 S a= 9 and b=1+4a sq rt u du which is 4(2/3)u^(3/2) a= 9 b = 1+4a

but when I do the substitution I get 265/8 = sq rt ((1+4a)^3) and since this is not a easy equation to solve I don't know what a is

Any idea where I went wrong???

Thanks

Struggling calculus beginner

2. $y = \frac{4}{3}x^{\frac{3}{2}}$

$y' = 2x^{\frac{1}{2}}$

$(y')^2 = 4x$

$\int_2^a \sqrt{1+4x} \, dx = \frac{49}{3}$

$\frac{1}{6}\left[(1+4x)^{\frac{3}{2}}\right]_2^a = \frac{49}{3}
$

$\left[(1+4x)^{\frac{3}{2}}\right]_2^a = 98
$

$(1+4a)^{\frac{3}{2}} - 27 = 98$

$(1+4a)^{\frac{3}{2}} = 125$

$1+4a = 125^{\frac{2}{3}}$

$1+4a = 25$

$a = 6$