6.1 & 6.4 an overview of the area problem area
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Transcript of 6.1 & 6.4 an overview of the area problem area
An Overview of the Area Problem
n = number of intervals arbitrary
Δx = size of intervals
i = which interval 1 to n
Finding the area could be broken down into the following parts:
Here, x1 = 0, n = 4, and Δx = 0.5
Lower estimate → left endpoints approximation method - LAM
Upper estimate → right endpoints approximation method - RAM
Here, x1 = 0, n = 4, and Δx = 0.5
Average estimate → mid endpoints approximation method -
MAM
Here, x1 = 0, n = 4, and Δx = 0.5
At some point we will use these formulas (pg.375 and pg. 376)
At some point you may want to use this formula:
How would we find the actual area?
Example Find the area of the region under the curve using: LAM
y = x2 [0, 2] n = 4
0 0.5 1 1.5 2
Example Find the area of the region under the curve using: RAM
y = x2 [0, 2] n = 4
0 0.5 1 1.5 2
Example Find the area of the region under the curve using: MAM
y = x2 [0, 2] n = 4
0.25 0.75 1.25 1.75
The Antiderivative method for Finding Areas
a x
A (x)
A’ (x) = f(x)
Let’s see how that worksExample: Find the area A(x) between the graph of f(x) = 2 and the interval [a, x] = [-1 ,x], and find the derivative A’(x) of this area function.
-1 x
y =2
2
x + 1
A = L* w
A = 2 * (x + 1)
A = 2 x + 2
A’ = 2 = f (x)
More Examples !!!Example: Find the area A(x) between the graph of f(x) = x + 1 and the interval [a, x] = [-1 ,x], and find the derivative A’(x) of this area function.
A =1/2 * b* h
A =1/2 * (x + 1) * (x + 1)
A =1/ 2 (x2 + 2x+ 1)
A = x2/2 + x + 1/2
y = x + 1
x + 1
-1 x x + 1
A’ = x + 1 = f (x)
One More Example !!!Example: Find the area A(x) between the graph of f(x) = 2x + 3 and the interval [a, x] = [-1 ,x], and find the derivative A’(x) of this area function.
A =1/2 *( b1 + b2) * h
A =1/2 *[ (2x + 3) + (2*(-1) + 3) ]* (x + 1)
A =1/2 * (2x + 4) * (x + 1)
A = 1/2 * (2x2 + 6x + 4)
A’ =2x + 3 = f (x)
y = 2x + 3
2x + 3
-1 x
x + 1
A = x2 + 3x + 2