Which expression correctly represents the volume of a solid formed by cross sections perpendicular to the x-axis with cross-sectional area A(x)?

Which expression correctly represents the volume of a solid formed by cross sections perpendicular to the x-axis with cross-sectional area A(x)?

• A. V = ∫_a^b (area of cross section) dx
• B. V = ∫_a^b (width of cross section) dx
• C. V = ∫_a^b (height of cross section) dx
• D. V = ∫_a^b (area of cross section) dx

Answer: (A)

The volume is found by summing up the volumes of thin slices whose thickness is dx. Since each slice perpendicular to the x-axis has cross-sectional area A(x), its volume is A(x) dx. Integrating from x = a to x = b adds up all those slice volumes, giving V = ∫_a^b A(x) dx. This matches how volume accumulates from area along the axis of stacking. The integral would not use a width or height alone, because those dimensions alone don’t capture the actual cross-sectional area that contributes to volume. If the cross sections were perpendicular to a different axis, you’d integrate with respect to that axis’s variable instead.

The volume is found by summing up the volumes of thin slices whose thickness is dx. Since each slice perpendicular to the x-axis has cross-sectional area A(x), its volume is A(x) dx. Integrating from x = a to x = b adds up all those slice volumes, giving V = ∫_a^b A(x) dx. This matches how volume accumulates from area along the axis of stacking. The integral would not use a width or height alone, because those dimensions alone don’t capture the actual cross-sectional area that contributes to volume. If the cross sections were perpendicular to a different axis, you’d integrate with respect to that axis’s variable instead.