Download Section 6.3 -‐ Area and the Definite Integral How do we find the a

Montana State University M161: Survey of Calculus 104 Section 6.3 -­‐ Area and the Definite Integral How do we find the area under a curve? We can try to approximate the area by placing rectangles under the curve. It is easier to find the area of rectangles and then add them together. Note: the lower images approximate the area under the curve better than the upper images. Def’n: Area under the Graph of a Function !"# ! ! !" ! !"!!#$%&'(# !"#$%#&"&' !"#$%&'# !" [!, !] Then the !"#! !" !"# !"#$%& !"#$% !"# !"#$% !" ! ! is: A = lim [! !! + ⋯ + ! !! ]Δ! !→!
!ℎ!"! !! , !! , … , !! !"# !"#$%"!"& !"#$%& !" !ℎ! ! !"#$%&'()*+! !" [!, !] !" !"#$% !"#$ℎ. The width of each rectangle is Δ! =
!−!
!
S. Schaefer Montana State University M161: Survey of Calculus 105 Riemann Sums Using Rectangles to Approximate Area under a Curve Step 1: Divide the interval [a, b] into n equal pieces. This is known as a partition. !!!
Step 2: Determine the width of each rectangle. Δ! = ! Step 3: Mark each point on the interval, including the endpoints. You should have n+1 marks on the interval [a, b] Step 4: Decide whether you are using: • LEFT endpoints • RIGHT endpoints • MIDPOINTS !!!
Step 5: Construct a series of rectangles, each rectangle has a base of Δ! = ! and a height based on the type of endpoint/midpoint Step 6: Find the height of each rectangle by calculating the function value. And use that to calculate the area of each rectangle. Step 7: Add the areas of the rectangles to approximate the area under the curve. S. Schaefer Montana State University M161: Survey of Calculus 106 Examples Find an approximation of the area under the graph of f(x) by computing the Riemann Sum of f(x) corresponding to the partition of the interval as shown in the accompanying figures. !"#! ≈ !"#! ≈ S. Schaefer Montana State University M161: Survey of Calculus 107 There has to be a better way than the rectangles! !"# !(!) = !" (A) Sketch the region R under the graph of f(x) on the interval [0, 2] and find its exact area using geometry. 1
1
!"#! = !ℎ = 2 6 = 6 2
2
(B) Use a Riemann Sum with 4 subintervals of equal length (n=4) to approximate the area of R. Choose the representative points to be the LEFT endpoints of the subintervals. Draw the rectangles. 2−0 1
Δ! =
= 4
2
1
3
1
!"#! ≈ ! 0 + !
+! 1 +!
2
2
2
= (C) Use a Riemann Sum with 8 subintervals of equal length (n=8) to approximate the area of R. Choose the representative points to be the LEFT endpoints of the subintervals. Draw the rectangles. 2−0 1
Δ! =
= 8
4
1
2
3
1
!"#! ≈ ! 0 + !
+!
+!
4
4
4
4
5
6
7
1
+ ! 1 +!
+!
+!
4
4
4
4
= (D) Compare the approximations obtained in parts (B) and (C) with the exact area found in part (A). Do the approximations get better when n gets larger? S. Schaefer Montana State University M161: Survey of Calculus 108 The Definite Integral and the Geometric Interpretation Definite Integral !"# ! ! !" !"#$%#&"&' !" !, ! !
!ℎ!" A = lim [! !! Δ! + ⋯ + ! !! Δ!] =
!→!
! ! !" !
This limit is called the definite integral of f(x) from a to b The number a is the lower limit of integration, And b is the upper limit of integration. Geometric Interpretation !" ! ! !" !"!!#$%&'(# !"# !"#$%#&"&' !" !, ! !
!"#$ !(!) !" !" !"#$% !" !ℎ! !"#! !" !ℎ! !"#$%& !"#$% !ℎ! !"#$ℎ !" ! ! !" [!, !] !
S. Schaefer