Question

True or False The volume of a solid of a known integrable cross section area \(A(x)\) from \(x=a\) to \(x=b\) is \(\int_{a}^{b} A(x) d x .\) Justify your answer.

Short answer

True, the statement is equivalent to the known formula in calculus for the volume of a solid with a known integrable cross-sectional area.

Step-by-step solution

Understand the problem

The statement suggests that the volume of a solid with a known cross-sectional area that is integrable can be found using the formula \(\int_{a}^{b} A(x) dx\). This is based on the concept of integration in calculus.

Compare with known formula

The known formula for the volume of a solid with cross-sectional area A(x) from x=a to x=b is indeed \(\int_{a}^{b} A(x) dx\). This formula essentially adds up all the infinitesimally small volumes of the slices between x=a and x=b.

Formulate answer

Since the statement is equivalent to the known formula in calculus for the volume of a solid with a known integrable cross-sectional area, the statement is true.