Chapter 1. calc_tutorial_13_3_003

1.1 Problem Statement

{3,4,5,6,7,8,9}
pow($a,2)-1

Compute the length of the curve over the given interval.

r(t) = <2t, ln t, t2>, 1 ≤ t ≤ $a

1.2 Step 1

Question Sequence

Question 1.1

Recall how to find the length of a path for a vector-valued function r(t) = <x(t), y(t), z(t)>. Assume that r(t) is differentiable and that r'(t) is continuous on [a, b]. Then the length s of the path r(t) for a t b is defined as follows.

We will need r'(t) to find the length of the curve defined by r(t) = <2t, ln t, t2> on the interval 1 ≤ t ≤ $a.

Find r'(t).

r'(t) =

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
Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Set up the integral that will give us the path length of r(t) = <2t, ln t, t2> on the interval 1 ≤ t ≤ $a.

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
Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Factor the radicand to simplify the integrand.

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

=

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
Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Integrate to find the length of the curve.

ifBsJAGAhCRIC/LhdDuAOmtghO2ZpRTqGZ+CWAkexa1bkWyXcfpFTYqSUfqrkYXFQjaLdVIQSK9tXAdDkTHzdVRmWFoMBMuCuXcMdZxMw9dX3s0fH+QV3LNrbDXY2ZjwVMUSmi4f5XHwXi7HfqPHslkZivIkQweST6Pn6qt5yQvr2Q1I/VD2pGyajCUuFPNIFKfPwWmt6RFDH5rD673Pg4gT1knorrzyCAYDYu8BmD0d/hd7dswx2lXH2M3xOOp4cxvv3HDIg7Y3Is3xoI9hlSZNw/b4ajxRa9MX4klElGFUjyWwyzkQc7GwAvXPGYsErWVaofPlVKJ+fYukfkkyZLHuksk/NqMOGgOlQhMLfh8ncIc2iFBKxBN3tCkByz+9hPgrtB1x+AP+G0vGReiJeMEeE5SpRx5ePUNSDrMQ0OVANbvrthts6+gCI8L1gF/kn6crrEgqMiqgB4ILAmcV/F+a5BXYU7e9wMcWjziP4S8eBSYYkAd1Sd+/l2UD6FE0k67r2MQeleKPocVwWEPvp7LQY3JvMlT1QpYmwCrKrWxpbX+3Iz263maVMls1PB/Yu9CdwCOb8MWf8QhgsuZb6eNXE9L1pNszCtDBz4clf+wr1z3dTVo0LZmH6OjXfPCZGfB5EgCNX6F8L4Gv+DbMnu7zba5ZwYsNSX7vfTXkXN23K+P16amm7mRK8AD/97SBqEquumrGU8ulX5ts8JLb/nXvlseoTsWCOo/WESMsblaI3S8EP5rivXmeM7sGeF4LauaFL4xMqahFdlsfMkY+D6VN2pK1AlFAsAbtaadXGU97eQWBefYoTJ3zj+RSi0VxqaZDBT6fAEo30pO8u+3Psly7af8oY06Nb3XbEXoTvQ9EU1R7hH55d8Bs2bNIHZhnhemyHwKh3wQgCli24kN9pK7rleKAyrSmrsINYBPlqIm0OO+d+iX3tMegLZKQJX9xtC1GVZhJatww61geyecQdU49f0GRbQ9787/p/0ozWzmKgYYeHr0/7QQvXmsF9G4TEimFMBeYZid1NidP8mIufoEbeloosSEKtMxuxiepPUuh3cNkR/YTHjfLB5zmR9IV3BOrUrIQ3Tnk+pandomXBWBsEwlnK8Of1QA6aL+vT8hVMQ2LYkuJWGNnm9zS8btwAZRQR7PmFLHYcsy3ThRCXZCpYSmLyxjgtc1Y2VOUe67fsxIv/Qet3smIpuWuntXkylPCKfL1yaMkbXgRMHj3hvjtHX6Z1ctyrC+MwNusuDAGHwCefTFUxOse+8G1e4Sci3qi5THrScq914C3

= iSba6t70dtA= + ln(nc1ItEz0kR4=)

Correct.
Incorrect.