Find the length of the curve.
-$$y = 2 x ^ { 3 / 2 } \text { from } x = 0 \text { to } x = \frac { 5 } { 4 }$$
A) $\frac { 335 } { 108 }$
B) $\frac { 9 } { 4 }$
C) $\frac { 335 } { 3 }$
D) $\frac { 335 } { 72 }$

Question

Quizplus · Accepted Answer

To find the length of the curve $y = 2x^{3/2}$ from $x = 0$ to $x = \frac{5}{4}$, we use the formula for the length of a curve in the form $y = f(x)$ over an interval $[a, b]$: $$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$First, find the derivative of $y = 2x^{3/2}$:$\frac{dy}{dx} = 2 \cdot \frac{3}{2}x^{1/2} = 3x^{1/2}$Then, substitute this into the formula:$$L = \int_{0}^{\frac{5}{4}} \sqrt{1 + (3x^{1/2})^2} dx = \int_{0}^{\frac{5}{4}} \sqrt{1 + 9x} dx$$Let's solve the integral:$$L = \int_{0}^{\frac{5}{4}} \sqrt{9x + 1} dx$$To solve this integral, we can use a substitution or recognize it directly as a standard form. The antiderivative of $\sqrt{9x + 1}$ can be found using standard techniques, leading to:$$L = \frac{1}{9} \left( \frac{2}{3}(9x + 1)^{3/2} \right) \Bigg|_{0}^{\frac{5}{4}}$$Evaluating this from $0$ to $\frac{5}{4}$:$$L = \frac{1}{9} \left( \frac{2}{3} \left( (9 \cdot \frac{5}{4} + 1)^{3/2} - (9 \cdot 0 + 1)^{3/2} \right) \right)$$$$L = \frac{1}{9} \left( \frac{2}{3} \left( (11.25 + 1)^{3/2} - 1^{3/2} \right) \right)$$$$L = \frac{1}{9} \left( \frac{2}{3} \left( 12.25^{3/2} - 1 \right) \right)$$$$L = \frac{1}{9} \left( \frac{2}{3} \cdot 335 \right)$$$$L = \frac{335}{108}$$Therefore, the length of the curve is $\frac{335}{108}$.

Find the Length of the Curve y=2x3/2 from x=0 to x=54y = 2 x ^ { 3 / 2 } \text { from } x = 0 \text { to } x = \frac { 5 } { 4 }y=2x3/2 from x=0 to x=45​

Find the Length of the Curve $y = 2 x ^ { 3 / 2 } \text { from } x = 0 \text { to } x = \frac { 5 } { 4 }$