Motion, area, and volume
What students will learn
Apply accumulation to motion, area, volume, and functional-curve length with justified signs, boundaries, and smooth pieces.
Before this topic
- Definite integrals and integration tools
- Curve intersections and geometric distance
Read and connect
Read: Finding the Average Value of a Function on an Interval →
Use an integral average and its units, rather than a secant rate or an unweighted mean of unequal-time samples.
Read: Connecting Position, Velocity, and Acceleration of Functions Using Integrals →
Recover position and velocity with initial values, and separate displacement from distance using velocity signs.
Read: Using Accumulation Functions and Definite Integrals in Applied Contexts →
Write initial amount plus accumulated net rate, then justify inventory values, extrema, and targets in context.
Read: Finding the Area Between Curves Expressed as Functions of x →
Find intersections and integrate a justified top-minus-bottom vertical strip.
Read: Finding the Area Between Curves Expressed as Functions of y →
Choose horizontal strips with correct branches, right-minus-left boundaries, and matching y-bounds.
Read: Finding the Area Between Curves That Intersect at More Than Two Points →
Split at relevant crossings so positive area does not disappear through signed cancellation.
Read: Volumes with Cross Sections: Squares and Rectangles →
Turn a stated square or rectangular cross section into an area function before integrating volume.
Read: Volumes with Cross Sections: Triangles and Semicircles →
Use the specified triangle, semicircle, or circular geometry, distinguishing a supplied radius from a diameter.
Read: Volume with Disc Method: Revolving Around the x- or y-Axis →
Choose perpendicular disc slices and measure a radius from a coordinate axis.
Read: Volume with Disc Method: Revolving Around Other Axes →
Measure disc radii as distances from the stated shifted axis, checking whether the region reaches it.
Read: Volume with Washer Method: Revolving Around the x- or y-Axis →
Identify nonnegative outer and inner radii and subtract their squared disc areas.
Read: Volume with Washer Method: Revolving Around Other Axes →
Measure both washer radii from the same shifted axis and keep the slice variable and bounds consistent.
Read: The Arc Length of a Smooth, Planar Curve and Distance Traveled →
Build functional-curve arc length from small distances, checking smoothness and splitting at corners.
Check understanding
Use the complete prompt to make student reasoning visible, then consult the separate worked reasoning and criteria. A reading completion tick records reading history.
Separate final position from total distance →
Add signed velocity accumulation to the initial position and split absolute-speed accumulation at a reversal.
Keep two area lobes from cancelling →
Find all crossings, justify the upper curve on each piece, and accumulate positive geometric area.
Measure washer radii from a shifted axis →
Choose the farther and nearer boundaries, square both distances separately, and integrate positive volume.
Find length across a corner →
Apply the smooth arc-length formula to each straight piece and explain why the corner requires a split.
Standards and earlier reading lists
These topic placements organize reading. Edition-qualified Standards mappings and the existing complete unit task sets retain their reviewed scope.