Integrate the area function $A(x)$ of the cross-sections over the interval $[a, b]$.

The volume is the accumulation of the areas of infinitely thin cross-sections.

Integration sums the areas of infinitely many infinitesimally thin slices to find the total volume.

The bounds define the region over which the volume is calculated; incorrect bounds lead to an incorrect volume.

Integrating with respect to x uses cross-sections perpendicular to the x-axis; integrating with respect to y uses cross-sections perpendicular to the y-axis.

Visualization helps determine the shape of the cross-sections, the limits of integration, and the correct formula for the area function.

They define the dimensions (side length, width, height) of the cross-sections.

Orientation determines whether to integrate with respect to x or y, and how the area function A(x) or A(y) is defined.

It involves integrating the difference between the upper and lower curves over a given interval.

Find the side length s, square it to get the area A(x), and integrate A(x) over the interval [a, b].

A two-dimensional shape formed by slicing a three-dimensional solid.

Area of the cross-section perpendicular to the x-axis at a given x.

A 3D solid formed by rotating a 2D shape around an axis.

Infinitesimally small thickness of the cross-section.

The limits of integration, defining the region's boundaries.

The amount of three-dimensional space occupied by an object.

Area of a square is $s^2$, where s is the side length.

Area of a rectangle is $w*h$, where w is the width and h is the height.

Intersecting at or forming right angles (90 degrees).

Integral evaluated between specific upper and lower limits, resulting in a numerical value.

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

$V = \int_a^b s^2 dx$

$V = \int_a^b w cdot h dx$

$A = s^2$

$A = w cdot h$

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

$w = f(y) - g(y)$

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

$V = \int_c^d A(y) dy$

$x = \sqrt[3]{y}$