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inna [77]
2 years ago
8

Find the length of the following curve. If you have a​ grapher, you may want to graph the curve to see what it looks like.

Mathematics
1 answer:
stepladder [879]2 years ago
7 0

The length of the curve y = \frac{1}{27}(9x^2 + 6)^\frac 32 from x = 3 to x = 6 is 192 units

<h3>How to determine the length of the curve?</h3>

The curve is given as:

y = \frac{1}{27}(9x^2 + 6)^\frac 32 from x = 3 to x = 6

Start by differentiating the curve function

y' = \frac 32 * \frac{1}{27}(9x^2 + 6)^\frac 12 * 18x

Evaluate

y' = x(9x^2 + 6)^\frac 12

The length of the curve is calculated using:

L =\int\limits^a_b {\sqrt{1 + y'^2}} \, dx

This gives

L =\int\limits^6_3 {\sqrt{1 + [x(9x^2 + 6)^\frac 12]^2}\ dx

Expand

L =\int\limits^6_3 {\sqrt{1 + x^2(9x^2 + 6)}\ dx

This gives

L =\int\limits^6_3 {\sqrt{9x^4 + 6x^2 + 1}\ dx

Express as a perfect square

L =\int\limits^6_3 {\sqrt{(3x^2 + 1)^2}\ dx

Evaluate the exponent

L =\int\limits^6_3 {3x^2 + 1} \ dx

Differentiate

L = x^3 + x|\limits^6_3

Expand

L = (6³ + 6) - (3³ + 3)

Evaluate

L = 192

Hence, the length of the curve is 192 units

Read more about curve lengths at:

brainly.com/question/14015568

#SPJ1

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