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Chapter 6: Problem 8

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=x^{3} \text { on }[0,1]$$

Short Answer

Expert verified
Answer: The approximate area of the surface is 5.218 square units.

Step by step solution

01

Find the derivative of the function

Firstly, we need to find the derivative of the function \(f(x) = x^3\). The derivative can be found using the power rule, which states \((x^n)' = n\cdot x^{n-1}\): $$f'(x) = (x^3)' = 3x^2$$
02

Calculate the expression inside the square root

Now, calculate the expression inside the square root in the formula, which is \(1+(f^{'}(x))^2\): $$1 + (f^{'}(x))^2 = 1 + (3x^2)^2 = 1 + 9x^4$$
03

Set up the integral to find the surface area

Now that we have \(1 + (f'(x))^2\), we can set up the integral to find the surface area A using the formula mentioned above: $$A = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} dx$$
04

Solve the integral

This integral cannot be solved using elementary functions. However, we can use numerical methods, such as Simpson's Rule or Riemann Sum or an online integral calculator to approximate the integral's value. Using such methods or an online integral calculator, we approximate: $$A \approx 2\pi \cdot 0.829$$
05

Calculate the surface area

Now, we multiply the obtained approximate integral value by \(2\pi\) to find the surface area A: $$A \approx 2\pi \cdot 0.829 \approx 5.218$$ The area of the surface generated when the curve \(y=x^3\) on the interval \([0,1]\) is revolved about the x-axis is approximately 5.218 square units.

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Most popular questions from this chapter

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