Calculate the volume of a solid building up from a defined base region.
Write integrals for different cross-sectional geometries (squares, semicircles, triangles).
Visualize how the base distance \( s = f(x) - g(x) \) translates to slice area \( A(x) \).
Key Equations
\( V = \int_a^b A(x) \, dx \)
Base distance: \( s = f(x) - g(x) \)
Square: \( A(x) = s^2 \)
Semicircle: \( A(x) = \frac{\pi}{8} s^2 \)
Eq. Triangle: \( A(x) = \frac{\sqrt{3}}{4} s^2 \)
Why It Matters
Not all 3D solids are cylindrical "solids of revolution." Knowing the cross-sectional geometry allows engineers to calculate exact volumes for irregular shapes such as architectural arches and biological tissues.
Tags
volumescross sectionsintegrationRiemann sumsAP Calculus AB
2D Base Region
Integrate \( \int A(x) \, dx \) =0.00
Approx \(\sum A(x_i) \Delta x\) =0.00
Base Region Functions
Cross Section Shape
Slices (Riemann N) 10
Drag the 3D model to rotate and view the solid from any angle. As slices N increases, it approaches the exact volume integral.