Approximating Area Under a Curve
Rectangles can be used to approximate the area under a curve.
Consider 𝑓(𝑥) = −0.1𝑥 2 + 10
Draw and shade 5 equal width rectangles under Draw and shade 5 equal width rectangles under
the curve with their upper right hand corner on the curve with their upper left hand corner on
the curve. Find the area of each rectangle.
the curve. Find the area of each rectangle.
Area from 5 right-hand rectangles =
Area from 5 left-hand rectangles =
Is this an over-estimate or under-estimate?
Is this an over-estimate or under-estimate?
10
10
8
8
6
6
4
4
2
2
5
10
5
10
What could be done with these areas to get a better estimate?
What could be done to the rectangles to give an even better estimate?
Area from 10 right-hand rectangles =
Area from 10 left-hand rectangles =
10
10
8
8
6
6
4
4
2
2
5
10
5
10
𝑥
0
1
2
3
4
5
6
7
8
9
10
𝑦
10
9.9
9.6
9.1
8.4
7.5
6.4
5.1
3.6
1.9
0
Trapezoids can also be used to approximate the area under a curve.
Draw and shade 5 equal width trapezoids
under the curve. Find the area of each
trapezoid.
Draw and shade 10 equal width trapezoids
under the curve. Find the area of each
trapezoid.
Area from 5 trapezoids =
Area from 10 trapezoids =
Is this an over-estimate or under-estimate?
Is this an over-estimate or under-estimate?
10
10
8
8
6
6
4
4
2
2
5
10
5
𝑥
0
1
2
3
4
5
6
7
8
9
10
𝑦
10
9.9
9.6
9.1
8.4
7.5
6.4
5.1
3.6
1.9
0
10
Is the trapezoid estimate a better or worse estimate than rectangles? Why?
Area units example: 𝐴 = 𝑏ℎ = miles
∙ hours = miles
hour
10
miles
hour
8
6
Area under velocity or speed curve is total distance traveled.
4
2
5
10
hours
Area Under Curve from Table Data Only:
Use the table to find area under the curve using 4 left hand rectangles.
𝑥
𝑦
0
1
2
15
4
21
6
25
8
29
𝑥
𝑦
0
1
2
15
4
21
6
25
8
29
𝐴 = 1(2) + 15(2) + 21(2) + 25(2) = 124
Use the table to find area under the curve using 4 right hand rectangles.
𝐴 = 15(2) + 21(2) + 25(2) + 29(2) = 180
Approximating Area Under a Curve
Rectangles can be used to approximate the area under a curve.
Consider 𝑓(𝑥) = −0.1𝑥 2 + 10
Draw and shade 5 equal width rectangles under Draw and shade 5 equal width rectangles under
the curve with their upper right hand corner on the curve with their upper left hand corner on
the curve. Find the area of each rectangle.
the curve. Find the area of each rectangle.
Area from 5 right-hand rectangles =
(2)(9.6) + (2)(8.4) + (2)(6.4) + (2)(3.6)
+ (2)(0) = 𝟓𝟔
Is this an over-estimate or under-estimate?
Area from 5 left-hand rectangles =
(2)(10) + (2)(9.6) + (2)(8.4) + (2)(6.4)
+ (2)(3.6) = 𝟕𝟔
Is this an over-estimate or under-estimate?
10
10
8
8
6
6
4
4
2
2
5
10
5
10
What could be done with these areas to get a better estimate?
Average the over- and under-estimates.
What could be done to the rectangles to give an even better estimate?
Make the rectangles thinner to have less over- or under-.
Area from 10 right-hand rectangles =
(1)(9.9) + (1)(9.6) + (1)(9.1) + (1)(8.4)
+(1)(7.5) + (1)(6.4) + (1)(5.1) + (1)(3.6)
+(1)(1.9) + (1)(0) = 𝟔𝟏. 𝟓
Area from 10 left-hand rectangles =
(1)(10) + (1)(9.9) + (1)(9.6) + (1)(9.1)
+(1)(8.4) + (1)(7.5) + (1)(6.4) + (1)(5.1)
+(1)(3.6) + (1)(1.9) = 𝟕𝟏. 𝟓
10
10
8
8
6
6
4
4
2
2
5
10
5
10
𝑥
0
1
2
3
4
5
6
7
8
9
10
𝑦
10
9.9
9.6
9.1
8.4
7.5
6.4
5.1
3.6
1.9
0
Trapezoids can also be used to approximate the area under a curve.
Draw and shade 5 equal width trapezoids
under the curve. Find the area of each
trapezoid.
Draw and shade 10 equal width trapezoids
under the curve. Find the area of each
trapezoid.
Area from 5 trapezoids =
(10 + 9.6)(2) (9.6 + 8.4)(2)
+
+
2
2
(8.4 + 6.4)(2) (6.4 + 3.6)(2)
+
+
2
2
(3.6 + 0)(2)
= 𝟔𝟔
2
Area from 10 trapezoids =
𝑥
0
1
2
3
4
5
6
7
8
9
10
(10 + 9.9)(1) (9.9 + 9.6)(1) (9.6 + 9.1)(1)
+
+
+
2
2
2
(9.1 + 8.4)(1) (8.4 + 7.5)(1) (7.5 + 6.4)(1)
+
+
+
2
2
2
(6.4 + 5.1)(1) (5.1 + 3.6)(1) (3.6 + 1.9)(1)
+
+
+
2
2
2
(1.9 + 0)(1)
= 𝟔𝟔. 𝟓
2
Is this an over-estimate or under-estimate?
Is this an over-estimate or under-estimate?
10
𝑦
10
9.9
9.6
9.1
8.4
7.5
6.4
5.1
3.6
1.9
0
10
8
8
6
6
4
4
2
2
5
10
5
10
Is the trapezoid estimate a better or worse estimate than rectangles? Why?
Much closer fit to curve.
Area units example: 𝐴 = 𝑏ℎ = miles
∙ hours = miles
hour
10
miles
hour
8
6
Area under velocity or speed curve is total distance traveled.
4
2
5
10
hours
Area Under Curve from Table Data Only:
Use the table to find area under the curve using 4 left hand rectangles.
𝑥
𝑦
0
1
2
15
4
21
6
25
8
29
𝑥
𝑦
0
1
2
15
4
21
6
25
8
29
𝐴 = 1(2) + 15(2) + 21(2) + 25(2) = 124
Use the table to find area under the curve using 4 right hand rectangles.
𝐴 = 15(2) + 21(2) + 25(2) + 29(2) = 180