Lesson 5: 8.5 Area Between Curves (with respect to y)
Duration of Days: 1
Lesson Objective
The student will be able to:
Determine when it is more efficient (or necessary) to integrate with respect to y rather than x.
Re-express functions given as y=f(x) into the form x=g(y) to facilitate horizontal partitioning.
Set up a definite integral using the "Right Curve minus Left Curve" method.
Identify limits of integration along the y-axis by finding the y-coordinates of the intersection points.
Why does "Top minus Bottom" fail or become overly complicated for certain regions (like those bounded by a sideways parabola)?
How do we determine the "height" and "width" of a representative rectangle when it is oriented horizontally?
How does the change in the variable of integration (dx to dy) affect the limits of the integral?
Horizontal Partitioning
Right Curve / Left Curve
Inverse Functions
y-limits of integration
Horizontal Representative Rectangle
CHA-5.A: Determine the area of a region bound by two or more curves which are functions of y.
Mathematical Practice 1.C: Identify an appropriate mathematical rule or procedure based on the classification of a given expression.
Mathematical Practice 4.C: Connect concepts to confirm or clarify the relationship between functions.
Description: This lesson teaches students to look at bounded regions "sideways." Instead of vertical rectangles, students use horizontal rectangles. The area is calculated by integrating the difference between the "Right-most" function and the "Left-most" function.
Purpose: Many regions are impossible to integrate with respect to x without splitting them into multiple parts (or at all, if the function isn't "one-to-one"). This lesson builds the flexibility needed for Volumes of Revolution (Disc/Washer Method) around the y-axis, which is a major component of the AP Exam.
DOK Levels
DOK Level 2 (Skill/Concept): Converting y=f(x) to x=g(y) and evaluating the resulting integral.
DOK Level 3 (Strategic Thinking): Comparing both x and y methods for a single region and justifying which is more efficient.
For Struggling Learners (Scaffolding):The "Head Tilt": Encourage students to physically tilt their heads 90 degrees to the right; the "Right" curve becomes the "Top" and the "Left" curve becomes the "Bottom."
Variable Consistency Check: Use a "Matching" drill where students must ensure that if the integral ends in dy, the functions inside must contain y and the limits must be y-values.
Color-Coding: Highlight the right boundary in one color and the left in another to clarify the subtraction order.
For Advanced Learners (Extension):Dual Integration: Challenge students to calculate the area of a specific region twice—once with respect to x and once with respect to y—to prove the results are identical.
Complex Boundaries: Provide a region where one boundary is a piecewise function, forcing a decision on which axis of integration is simpler.
AP College Board Assessment