Math Problem Statement

Find the area of the surface generated when the given curve is revolved about the given axis.

yequals=x cubedx3​,

for

0 less than or equals x less than or equals 10≤x≤1​;

about the​ x-axis

Solution

To find the area of the surface generated when the curve y=x3y = x^3 is revolved about the xx-axis over the interval 0x10 \leq x \leq 1, we use the formula for the surface area of a solid of revolution:

A=2πaby1+(dydx)2dxA = 2\pi \int_{a}^{b} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx

Step 1: Find dydx\frac{dy}{dx}

Given y=x3y = x^3, we first compute the derivative dydx\frac{dy}{dx}:

dydx=3x2\frac{dy}{dx} = 3x^2

Step 2: Plug yy and dydx\frac{dy}{dx} into the formula

Substitute y=x3y = x^3 and dydx=3x2\frac{dy}{dx} = 3x^2 into the surface area formula:

A=2π01x31+(3x2)2dxA = 2\pi \int_{0}^{1} x^3 \sqrt{1 + \left(3x^2\right)^2} \, dx

Simplify the expression inside the square root:

1+(3x2)2=1+9x41 + \left(3x^2\right)^2 = 1 + 9x^4

So the integral becomes:

A=2π01x31+9x4dxA = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx

Step 3: Evaluate the integral

The integral 01x31+9x4dx\int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx can be challenging to solve analytically, but it can be handled using substitution or numerical methods.

Numerical Evaluation

Given the complexity, we might use numerical integration techniques to approximate the value of the integral. However, I can set up the integral here:

A=2π01x31+9x4dxA = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx

The exact value of this integral would give us the area of the surface. The integral typically does not have a simple antiderivative, so evaluating it numerically is the most practical approach.

Final Answer:

The surface area is given by:

A=2π01x31+9x4dxA = 2\pi \int_{0}^{1} x^3 \sqrt{1 + 9x^4} \, dx

Would you like to go through the numerical approximation or discuss any particular method to evaluate this integral?


Related Questions:

  1. How does the surface area change if the curve is revolved around the yy-axis instead of the xx-axis?
  2. What would be the surface area if the curve were y=x2y = x^2 instead of y=x3y = x^3?
  3. Can this integral be solved using a substitution method? If so, how?
  4. How can numerical methods such as Simpson's rule or the trapezoidal rule be applied to estimate this integral?
  5. What is the geometric significance of the term 1+(dydx)2\sqrt{1 + \left(\frac{dy}{dx}\right)^2} in the surface area formula?

Tip: For surface area problems, setting up the integral correctly is crucial. If the integral is difficult to evaluate analytically, consider using numerical methods for an approximation.

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Math Problem Analysis

Mathematical Concepts

Calculus
Integral Calculus
Surface Area of Revolution

Formulas

Surface area of revolution formula

Theorems

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Suitable Grade Level

College Level