Lecture 2: Volume By Cross Section

Take a solid, and imagine that you "chop it up" along the x,y, or z-axis into "small" slices of thickness d(x) (or d(y), or d(z), depending on the axis)

The volume of that particular slice is then

Volume = (cross-sectional area)·(thickness) = A(z) dz

Why? Because it looks like a cylinder.

This is just one slice: To get the volume of the entire solid, we have to add up (= integrate) all the slices:

Calculating volumes by cross-sections :

• If a solid lies between a≤x≤b, and cross-sections perpendicular to the x-axis have area A(x), then its volume can be calculated from

• If a solid lies between c≤y≤d, and cross-sections perpendicular to the y-axis have area A(y), then its volume can be calculated from

• If a solid lies between m≤z≤n, and cross-sections perpendicular to the z-axis have area A(z), then its volume can be calculated from

Comments: For each problem, we will need to determine the expression for A(x) (or A(y), A(z)).

Typically, cross-sections will be a standard geometrical shape (like a square, rectangle, triangle, circle/semi-circle) since we know the areas of these shapes.

Common formulas for A(x): (change as necessary for A(y) or A(z); not an exhaustive list)

• Cross-sections are squares:

A(x)=ℓ*(x)2 where ℓ(x)=(y_top)(x)−(y_bot)(x)

• Cross-sections are semi-circles:

A(x)=1/2*π*(d(x)/2)2 where d(x)=(y_t*o*p)(x)−(y_b*o*t)(x)

• Cross-sections are equilateral triangles:

A(x)=√(,3)/4*ℓ*(x)2 where ℓ(x)=(y_t*o*p)(x)−(y_b*o*t)(x)

Example: A solid has base in the xy-plane given by the region enclosed between y=1, x=tan((π*y)/6), and the y-axis, as shown.

Find the volume of the solid if cross-sections perpendicular to the y-axis are squares whose bases run from the y-axis to x=tan((π*y)/6).

Soln: