Area Between Curves

Top minus bottom, integrated.

12 problems · about 22 minutes · CHA-5.A

What this lesson teaches

The student computes the area between two curves and interprets accumulation in context.

The area between curves is the integral of top minus bottom.
Set the functions equal to find the limits, then test a point to see which is on top.
When two rates are graphed, the area between them is the net accumulation.

Straightforward practice. Get the method working first.

5 problems

y = 7 and y = 2 from x = 0 to x = 4. What is the area between them?
Answer 20
Why 20.
What is the integrand for an area between curves?
Answer The upper function minus the lower one.
Why Top minus bottom.
y = 4 and y = 1, from x = 0 to x = 5. What is the area between them?
Answer 15
Why A rectangle 3 tall and 5 wide.
y = x² and y = 0 from 0 to 3. Antiderivative x³/3. Area?
Answer 9
Why 27 ÷ 3.
y = x and y = x² cross at x = 0 and x = 1. On (0, 1), which is on top? 1 the line, 2 the parabola.
Answer 1
Why Test x = 0.5.

The same ideas with more to keep track of.

3 problems

Flow in 10 L/min, out 4 L/min, for 6 minutes. Net gain in litres?
Answer 36
Why 6 L/min for 6 minutes.
y = 6 and y = 2 from x = 1 to x = 4. Area between?
Answer 12
Why 4 tall and 3 wide.
If the curves cross inside the region, how many integrals are needed at minimum?
Answer 2
Why One on each side of the crossing.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the area-between-curves method in order.
Answer 1. Set the two functions equal to find the crossings. 2. Test a point inside to see which is on top. 3. Integrate top minus bottom between the crossings. 4. Split the integral if the curves trade places.
Why The crossings must be known before anything can be tested.
y = 9 and y = 4 from x = 0 to x = 2. Area between?
Answer 10
Why 5 tall and 2 wide.
The Crossings: y = x and y = x² cross at x = 0 and which other x?
Answer 1
Why x = 1.
The Tank: Water flows in at 8 L/min and out at 3 L/min for 10 minutes. What is the net gain, in litres?
Answer 50 L
Why 50 L.