Cogito
AP Calculus AB · Chapter 8 · Lesson 1
Area Between Curves
Top minus bottom, every time.
11 problems · about 24 minutes · AP Calculus AB 8.4, 8.5
What this lesson teaches
The student computes the area between two curves over an interval.
- The area between curves is the integral of the top function minus the bottom one.
- The limits are usually where the curves cross.
- Split the integral wherever they swap places, and expect a positive answer.
Warm Up
Straightforward practice. Get the method working first.
4 problemsy = 5 and y = 1 from 0 to 4. What is the area?
Answer 16
Why 16.
y = 4 and y = 0 from 0 to 3. What is the area?
Answer 12
Why A rectangle.
y = x and y = 0 from 0 to 4. What is the area?
Answer 8
Why A triangle.
y = 6 and y = 2 from 0 to 5. What is the area?
Answer 20
Why Height 4, width 5.
Build It Up
The same ideas with more to keep track of.
3 problemsA negative answer for an area means what? 1 for the order was wrong, 2 for the area is negative.
Answer 1
Why Area is never negative.
y = x² and y = 0 from 0 to 3. What is the area?
Answer 9
Why x³/3 at 3.
If two curves cross inside the interval, must you split the integral? 1 for yes, 0 for no.
Answer 1
Why The top function changes.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsy = 2x and y = 0 from 0 to 3. What is the area?
Answer 9
Why x² at 3.
y = 10 and y = 4 from 1 to 3. What is the area?
Answer 12
Why Height 6, width 2.
The Crossings: y = x and y = x² cross at x = 0 and at which other x?
Answer 1
Why x = 1.
Which Is On Top: Between 0 and 1, which is greater at x = 0.5? 1 for x, 2 for x².
Answer 1
Why The line.