Area under the curve
Area under the curve
April 13, 2018
Area
Objectives Understand the concept of area Approximate the area of a plane region Find the area of a plane region using limits
1
Area under the curve
April 13, 2018
Sometimes in Calculus, you want to estimate the area of a plane region. There are several different ways to do this: Rectangles Midpoint Rule Riemann Sums Trapezoidal Rule
Left and Right Endpoints
2
Area under the curve
April 13, 2018
To find the area under a curve, we approximate the area using rectangles and then use limits to find the area.
Estimate the area under the curve of y = 1 x2, 0 ≤ x ≤ 1, using 4 rectangles.
The first thing you want to do is determine the width of each rectangle. You do this by using the formula:
Where b is the highest x value and a is the lowest and n is the number of rectangles.
3
Area under the curve
April 13, 2018
So, the intervals that you will use will be the following: [0, 1/4], [1/4, 1/2], [1/2, 3/4], [3/4, 1]
Now we need to label each of the end points.
4
Area under the curve
April 13, 2018
Next, we evaluate each of the endpoints.
x0
x1
x2
x3
x y=1x2
To find the estimate, we now have to find the summation of the left endpoints and the summation of the right endpoints.
Left endpoints:
5
Area under the curve
April 13, 2018
x
x0 0
y=1x2
1
x1 1/4
x2 1/2
x3 3/4
x4 1
15/16 3/4 7/16
0
Right endpoints:
6
Area under the curve
April 13, 2018
x
x0 0
y=1x2
1
x1 1/4
x2 1/2
x3 3/4
x4 1
15/16 3/4 7/16
0
The estimate of the area under y = 1 x2, from 0 to 1 with an n of 4 is: .53125 ≤ x ≤ .78125
The actual area under the curve is:
7
Area under the curve
April 13, 2018
8
Area under the curve
April 13, 2018
Midpoint Rule
For the Midpoint rule, you must use the same intervals, but the xvalues will change.
[0, 1/4], [1/4, 1/2], [1/2, 3/4], [3/4, 1]
Midpoints: 1/8, 3/8, 5/8, 7/8
9
Area under the curve
April 13, 2018
x
x0 1/8
x1 3/8
x2 5/8
x3 7/8
y=1x2
Midpoint:
10
Area under the curve
April 13, 2018
x
y=1x2
x0
x1
x2
x3
1/8
3/8
5/8
7/8
63/64 55/64 39/64 15/64
11
Area under the curve
April 13, 2018
12
April 13, 2018
Area
Objectives Understand the concept of area Approximate the area of a plane region Find the area of a plane region using limits
1
Area under the curve
April 13, 2018
Sometimes in Calculus, you want to estimate the area of a plane region. There are several different ways to do this: Rectangles Midpoint Rule Riemann Sums Trapezoidal Rule
Left and Right Endpoints
2
Area under the curve
April 13, 2018
To find the area under a curve, we approximate the area using rectangles and then use limits to find the area.
Estimate the area under the curve of y = 1 x2, 0 ≤ x ≤ 1, using 4 rectangles.
The first thing you want to do is determine the width of each rectangle. You do this by using the formula:
Where b is the highest x value and a is the lowest and n is the number of rectangles.
3
Area under the curve
April 13, 2018
So, the intervals that you will use will be the following: [0, 1/4], [1/4, 1/2], [1/2, 3/4], [3/4, 1]
Now we need to label each of the end points.
4
Area under the curve
April 13, 2018
Next, we evaluate each of the endpoints.
x0
x1
x2
x3
x y=1x2
To find the estimate, we now have to find the summation of the left endpoints and the summation of the right endpoints.
Left endpoints:
5
Area under the curve
April 13, 2018
x
x0 0
y=1x2
1
x1 1/4
x2 1/2
x3 3/4
x4 1
15/16 3/4 7/16
0
Right endpoints:
6
Area under the curve
April 13, 2018
x
x0 0
y=1x2
1
x1 1/4
x2 1/2
x3 3/4
x4 1
15/16 3/4 7/16
0
The estimate of the area under y = 1 x2, from 0 to 1 with an n of 4 is: .53125 ≤ x ≤ .78125
The actual area under the curve is:
7
Area under the curve
April 13, 2018
8
Area under the curve
April 13, 2018
Midpoint Rule
For the Midpoint rule, you must use the same intervals, but the xvalues will change.
[0, 1/4], [1/4, 1/2], [1/2, 3/4], [3/4, 1]
Midpoints: 1/8, 3/8, 5/8, 7/8
9
Area under the curve
April 13, 2018
x
x0 1/8
x1 3/8
x2 5/8
x3 7/8
y=1x2
Midpoint:
10
Area under the curve
April 13, 2018
x
y=1x2
x0
x1
x2
x3
1/8
3/8
5/8
7/8
63/64 55/64 39/64 15/64
11
Area under the curve
April 13, 2018
12
