$V = \int_a^b A(x),dx$

$A = \frac{\sqrt{3}}{4}s^2$

$V = \int_a^b \frac{\sqrt{3}}{4}s^2 ,dx$

$A = \frac{1}{2}s^2$

$V = \int_a^b \frac{1}{2}s^2 ,dx$

$A = \frac{1}{2}\pi r^2$

$V = \int_a^b \frac{1}{2}\pi r^2 ,dx$

$s = h(x) - g(x)$

$r = \frac{h(x) - g(x)}{2}$

Set $h(x) = g(x)$ and solve for x.

1. Find the side length s in terms of x. 2. Determine the bounds of integration [a, b]. 3. Set up the integral: $V = \int_a^b \frac{\sqrt{3}}{4}s^2 ,dx$. 4. Evaluate the integral.

1. Find the radius r in terms of x. 2. Determine the bounds of integration [a, b]. 3. Set up the integral: $V = \int_a^b \frac{1}{2}\pi r^2 ,dx$. 4. Evaluate the integral.

1. Identify the upper function h(x) and the lower function g(x). 2. Calculate $s = h(x) - g(x)$.

1. Identify the upper function h(x) and the lower function g(x). 2. Calculate $r = \frac{h(x) - g(x)}{2}$.

1. Set the two bounding functions equal to each other: $h(x) = g(x)$. 2. Solve for x to find the intersection points.

1. Determine the area function A(x) based on the cross-section shape. 2. Identify the bounds of integration [a, b]. 3. Write the integral: $V = \int_a^b A(x) ,dx$.

1. Find the antiderivative of A(x). 2. Evaluate the antiderivative at the upper and lower bounds. 3. Subtract the value at the lower bound from the value at the upper bound.

1. Simplify the integrand. 2. Use u-substitution if possible. 3. Apply integration techniques such as integration by parts if needed.

1. Ensure the units are correct (cubic units for volume). 2. Verify that the volume is positive. 3. Use a calculator or software to check the integral calculation.

1. Express the bounding functions in terms of y. 2. Integrate with respect to y: $V = \int_c^d A(y) ,dy$.

The integral of the area function A(x) from a to b: $V = \int_a^b A(x),dx$

A function representing the area of a cross-section perpendicular to the x-axis.

The length of a side of the triangle that forms the cross-section, often determined by the difference between two functions.

The radius of the semicircle, often half the distance between two bounding functions.

The x-values that define the interval over which the solid exists, often found by intersection points of bounding curves.

A 2D shape obtained by slicing a 3D object.

The amount of space occupied by a three-dimensional object or region of space, expressed in cubic units.

A triangle in which all three sides have the same length.

A triangle that has a right angle and two sides of equal length.

A half-circle, formed by cutting a circle in half through its center.