1. Application: Area under and between Curve(s) Volume Generated
Integration: Part 3
Application: Area under and between Curve(s)
Volume Generated

2. Integration: Application
Area
Volume
Work
Pressure

3. Definite Integration: Refresh
Use your calculator to double check your answers

4. Application : Area Under a Curve
Shaded area that is bordered by y=f(x), x=a, x=b and x-axis is
Area above x-axis is +ve
Area below x-axis is –ve and would need to |-ve| y=
x=a
x=b A

5. Application: Area Under a Curve

6. Application: Area Under a Curve

7. Application: Area Under a Curve - Steps
Step-by-Step
Identify the function.
Sketch the graph to visualise (if needed)
Visualise and shade the area in question
Identify the border(s) for the area
Perform definite Integration, accordingly.
If –ve prediction, absolute the value using the | | sign
Add together the area (s) (if needed)
Note in unit2 (An are MUST be +ve)

8. Area Under a Curve: Example

9. Application: Area Between Curves
If f and g are continuous with f(x) => g(x) throughout [a , b], then the area of the region between the curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b.

10. Area Between Curves

11. Area Between Curves - Steps
Step 1: Sketch the curves and note the intersecting points
Step 2: Find the limits of integration by finding the intersecting points (y = y).
Step 3: Write a formula for f(x) – g(x) (depending on the which is the top curve and bottom curve). Simplify it.
Step 4: Integrate f(x) – g(x) of Step3 from a to b. The value obtained is the area (units2).

12. Intersecting curves exampley 2 -1 1 -2 3
y = 2 - x2
y = - x x

13. Areas between curves
The region runs from x = -1 to x = 2. The limits of integration are a = -1, b =2. The area between the curves is

14. Area Between Curves Find area enclosed between y= - x2 + 5x and y=2x
Find area between y = x 2 - 2x + 2 and y=-x 2 + 6
Find area between y = x 2 – 2x+ 3 and y = 2x3 -12x

15. Non-intersecting curves example
/4 1
y = sin x
y = sec2x
x-axis 2
(x, g(x))
y-axis
(x, f(x)) x

16. Area between a curve and a line (trigonometric function)
y =1 
y =sin2 x
/2
x-axis
y-axis 1

17. Area Between Curves Find area between y=x2 and y=-x2 [3 , 6]
Find area enclosed between y= - x2 + 5x and y=8x+10 [-1 ,0]
Find area between y = sin(x) and y=1 [3 , 6]
Find area between y = cos(x) and y=-1[-8 , -7]

18. Integration with respect to y-axis
If a region’s bounding curves are described by functions of y, are would be easier calculated horizontal instead of vertical and the basic formula has y in place of x
Area under the curve, you would need to modify the equation to be in terms of y
Formula for Area between curves

19. Integration with respect to y-axis
Find the area that is bounded by x = y + 2 , x = y2 and by the x-axis.

20. Integration with respect to y-axis
The region’s right-hand boundary is the line
x = y + 2, so f (y) = y + 2
The left-hand boundary is the curve x = y2,
so g (y) = y2.
The lower limit of integration is y = 0.
We find the upper limit by solving x = y + 2 and
x = y2 simultaneously for y:
y = -1, y = 2

21. Y-axis: Examples
Find area under the curve for x = 8y – y2 from y = 0 to y = 7
Find area between x=y2 and y=2x – 1
Find area enclosed between x = (y - 2)2 and x=1
Find area between x = 4y – 2y2, y=2x - 1

22. Application: Volume Generated (Disc Method)

23. Volume Generated - Steps
STEP 1: Square the equation i.e. (3x)2
STEP2: Perform steps like Area under/between curves
STEP 2.5: Definite Integral as such
STEP 3: State in units3

24. Application: Volume Generated (2 Curves)

25. Volume Generated: Examples
Find volume generated from the curve y = x3 + x2 – 6x [-3 , 0] rotated along the x-axis.
Find volume generated between y = 2x2 - 4x + 6 and y = x2 + 2x + 1 for in respect to the x-axis
Find volume generated between y=x2 - 2x +1 and y=2x – 1 being rotated along the x-axis
Find volume generated between y = - x2 + 5x and y = 2x rotated along the x-axis