65. Volume Find the volume of the solid generated when the region bounded by y = sin²(x) * cos^(3/2)(x) and the x-axis on the interval [0, π/2] is revolved about the x-axis.

Verified step by step guidance

1. Step 1: Recognize that the problem involves finding the volume of a solid of revolution. The formula for the volume when a region is revolved about the x-axis is given by: $V\; =\; \pi \; \int [a,b]\; f(x)²\; dx$, where f(x) is the function describing the region and [a, b] is the interval of integration.
2. Step 2: Identify the function and interval from the problem. Here, the function is $f(x)\; =\; sin²(x)\; *\; cos^(3/2)(x)$, and the interval is $[0,\; \pi /2]$. Substitute these into the volume formula.
3. Step 3: Set up the integral for the volume. The integral becomes: $V\; =\; \pi \; \int [0,\; \pi /2]\; (sin²(x)\; *\; cos^(3/2)(x))²\; dx$. Simplify the integrand by squaring the function: $(sin²(x)\; *\; cos^(3/2)(x))²\; =\; sin⁴(x)\; *\; cos³(x)$.
4. Step 4: Rewrite the integral with the simplified integrand: $V\; =\; \pi \; \int [0,\; \pi /2]\; sin⁴(x)\; *\; cos³(x)\; dx$. To solve this integral, consider using trigonometric identities or substitution methods. For example, you might use the identity $sin²(x)\; =\; 1\; -\; cos²(x)$ to simplify further.
5. Step 5: Apply appropriate integration techniques, such as substitution or reduction formulas, to evaluate the integral. Once the integral is solved, multiply the result by π to find the volume. Ensure all steps are carefully followed to handle the trigonometric powers correctly.

Key Concepts

Volume of Revolution

The volume of revolution refers to the volume of a solid formed by rotating a two-dimensional shape around an axis. In calculus, this is typically calculated using methods such as the disk method or the washer method, which involve integrating the area of circular cross-sections perpendicular to the axis of rotation.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area or volume, over a specific interval. In this context, it is used to find the volume of the solid generated by revolving the given function around the x-axis, integrating from the lower to the upper bounds of the interval, [0, π/2].

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with periodic phenomena. The function y = sin²(x) * cos^(3/2)(x) combines these functions, and understanding their behavior and properties is essential for evaluating the integral and determining the volume of the solid formed.