1. Application of the Integral
Section 4-2 (a)
Application of the Integral

2. 1) The graph on the right, is of the equation
How would you find the area of the shaded region?

3. 2) The graph on the right, is of the equation of a semicircle
How would you find the
area of the shaded region?

4. Area of Common known Geometric shapes
Triangle –
Rectangle –
Semicircle –
Trapezoid –
*Give exact area under the curve

5. What if the curve doesn’t form a geometric shape?
Determine area is by finding the sum of rectangles
Use rectangles to approximate the area between the curve and the x – axis: Archimedes (212 BC)
Add the area of the rectangles to approximate the area under the curve
Each rectangle has a height f(x) and a width dx

6. Consider the equation:

7. How can we get a better approximation?
3) find the area under the curve
from x = 1 to x = 5 using two rectangles of
equal width. 1 5 3
How can we get a better approximation?

8. 4) For the previous problem use four rectangles
More rectangles
4) For the previous problem use four rectangles

9. Even More Rectangles
Suppose we increase the number of rectangles, then the area underestimated by the rectangles decreases and we have a better approximation of the actual area.
How can we get an even better approximation?

10. Rectangles formed by the left-endpoints:
Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the left most endpoint is given by:
Value of function at a
the leftmost endpoint
Value of function at
second to last endpoint .
Excludes the rightmost
endpoint
Values of function at
intermediate x-values
Width of each rectangle
along the x-axis

11. Left end-point rectangles
The sum of the areas of the rectangles shown above is called a left-hand Riemann sum because the left-hand corner of each rectangle is on the curve.

12. Rectangles formed by the right-endpoints:
Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the right most endpoint is given by:
Value of function at
2nd x-value. Excludes
the leftmost endpoint
Value of function at the
last endpoint b.
Width of each rectangle
along the x-axis
Values of function at
intermediate x-values

13. Right end-point rectangles
The sum of the areas of the rectangles shown above is called a right-hand Riemann sum because the right-hand corner of each rectangle is on the curve.

14. Circumscribed vs. Inscribed
Circumscribed Rectangles: Extend over the curve and over estimate the area
Inscribed Rectangle: Remain below the curve and under estimate the area

15. Upper and Lower Sums Upper Sum The sum of the circumscribed rectangles
The sum of the inscribed rectangles

16. 5) Approximate the area under the curve
on the interval [ 0 ,4] and n = 4

17. 6) Find the upper and lower sums of
on the interval [ 0 ,3] and n = 3

18. Left-Endpoint Approximations
Circumscribed: when the function
is decreasing
Inscribed: when the function is increasing
Right-Endpoint Approximations
is increasing
Inscribed: when the function is decreasing

19. 7) Use left endpoints to approximate the area under the curve
on the interval [ 0 ,3] and n = 3

20. 8) Use right endpoints to approximate area under
on the interval [ 0 ,2] using 8 rectangles

21. 9) Use left endpoints to approximate area under
on the interval [ 0 ,2] using 8 rectangles

22. Homework
Page 268 # 25, 26, 27, 29, 31, 33, 34, 35, 41 and 43

23. Homework
Page 268 # 34