Skip to main content
CalculusBy Educator Today

C35 · Unit C8 Applications of integration

Area between curves with x or y slices

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.4–8.6
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Area between curves uses a nonnegative slice length. For vertical slices integrate upper minus lower with respect to x; for horizontal slices integrate right minus left with respect to y. Find intersections first and split wherever order changes. Choosing the other orientation can reduce the number of pieces.

Worked example

Between y equals x and y equals x squared on [0,1], the line lies above the parabola. Area equals ∫ sub 0 to the power 1 (x minus x squared)dx equals 1 over 2 minus 1 over 3 equals 1 over 6. In horizontal slices, the right boundary is x equals square root of y and left is x equals y, giving ∫ sub 0 to the power 1 ( square root of y minus y)dy equals 1 over 6.

Explore the mathematics

Area between curves with x or y slices

Find intersections and integrate the positive height between curves.

Setting 3 of 5

Scroll sideways to see the full diagram.

Where do the curves intersect, and which is above?xy1Marked input 0.5; displayed curve height 0.25Between y=x and y=x² on [0,1]

Predict before continuing

Where do the curves intersect, and which is above?

Read the complete explanation
  1. Where do the curves intersect, and which is above?
  2. Solve x=x²: intersections x=0 and x=1.
  3. On this interval the vertical slice height is x−x².
  4. Area ∫₀¹(x−x²)dx=1/6. If the order changes, split the interval.
See the idea: Area between curves with x or y slices
See the idea: Area between curves with x or y slicesThe region lies between the upper line y=x and lower parabola y=x² on [0,1]. A vertical slice has width w=x−x². Integrating w gives area 1/6; horizontal slices use √y−y.xy000.250.250.50.50.750.7511

The region lies between the upper line y=x and lower parabola y=x² on [0,1]. A vertical slice has width w=x−x². Integrating w gives area 1/6; horizontal slices use √y−y.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Integrating one difference across a crossing may cancel area. Determine the ordering interval by interval.

Investigate and explain

Solve one region using both vertical and horizontal slices. Compare the boundaries and limits carefully.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.