Question

question 4
4 pts
in the formula for the volume of a solid, ( v=int_{a}^{b}a(x)dx ), what best defines ( a(x) )?
( a(x) ) is the surface area of the solid
( a(x) ) is the cross - sectional area at a given x
the correct answer is not listed.
( a(x) ) is the area of the base of the solid
( a(x) ) is the maximum value of the solid in a given section

Explanation:

The formula \(V=\int_{a}^{b}A(x)dx\) is used to find the volume of a solid of revolution (or a general solid with known cross - sectional area). Here, the integral sums up the infinitesimal volumes \(A(x)dx\). The term \(A(x)\) represents the area of the cross - section of the solid at a particular value of \(x\).

• The surface area of the solid is a different quantity and is not related to this volume formula in the way \(A(x)\) is defined here.
• The area of the base is a single - valued area (for a solid like a prism with a constant base, but the formula \(\int_{a}^{b}A(x)dx\) is more general for non - constant cross - sections).
• The maximum value of the solid in a given section is not an area measure and has no relation to the volume formula \(V = \int_{a}^{b}A(x)dx\).

Answer:

\(A(x)\) is the cross - sectional area at a given \(x\)