Area Between Curves

Top minus bottom, integrated.

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

NameDate

Figure — use these to answer the problems

The shaded strip is bounded above and below. Between two curves, each strip has height top minus bottom.

y = 1x² + 0x + 0

Left limit-1

Right limit2

Figure 1

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
2
What is the integrand for an area between curves?
- The upper function minus the lower one.
- The two functions added together.
y = 4 and y = 1, from x = 0 to x = 5. What is the area between them?
Answer
y = x² and y = 0 from 0 to 3. Antiderivative x³/3. Area?
Answer
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

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
y = 6 and y = 2 from x = 1 to x = 4. Area between?
Answer
If the curves cross inside the region, how many integrals are needed at minimum?
Answer

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

4 problems

9
Put the area-between-curves method in order.
Write 1 to 4 in the boxes to put these in order.
- Set the two functions equal to find the crossings.
- Test a point inside to see which is on top.
- Integrate top minus bottom between the crossings.
- Split the integral if the curves trade places.
y = 9 and y = 4 from x = 0 to x = 2. Area between?
Answer
The Crossings
y = x and y = x² cross at x = 0 and which other x?
Answer
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?
AnswerL