Volume Formulas AP Calc

Volume Formulas:

\[ V = \int_{a}^{b} \pi \times [f(x)]^2 \,dx \text{ (disk method)} \] \[ V = \int_{a}^{b} 2\pi \times x \times f(x) \,dx \text{ (shell method)} \]

Volume (cubic units):

Definition: These formulas calculate volumes of solids of revolution using integration techniques.

Purpose: They help determine volumes of complex shapes generated by rotating a function around an axis.

The calculator uses two methods:

\[ V = \int_{a}^{b} \pi \times [f(x)]^2 \,dx \text{ (disk method)} \] \[ V = \int_{a}^{b} 2\pi \times x \times f(x) \,dx \text{ (shell method)} \]

Where:

Explanation: The disk method sums circular cross-sections, while the shell method sums cylindrical shells.

Details: These techniques are fundamental in calculus and have applications in physics, engineering, and 3D modeling.

Tips: Select method, enter function (e.g., "x^2", "sin(x)"), and integration limits. The calculator will evaluate the integral.

Q1: When to use disk vs. shell method?
A: Disk method is typically used when rotating around a horizontal axis, shell method for vertical axes.

Q2: What functions can I enter?
A: Most standard functions: polynomials, trig, exponential, logarithmic functions.

Q3: What if I get a negative volume?
A: Volume is always positive. The calculator takes absolute value of the integral.

Q4: How accurate are the results?
A: Results are mathematically exact (within computational limits).

Q5: Can I use this for non-revolution volumes?
A: No, these formulas specifically calculate volumes of revolution.