Introduction to Integration – Area Under The Curve Using Trapezoids

In the previous article, we learned how to approximate the area under curves using rectangle Riemann sums. In this article, we will estimate the area using trapezoids.

For example, let’s say you were asked to estimate the area of the following curve y = 1- x2 on the interval of [0,1]:

Of course, we can estimate the area using rectangular Riemann sums, but our approximations won’t be precise. Therefore, we are going to use trapezoids. Let’s compare the difference using 2, 4, 6, and 10 partitions:

2 Partitions


4 Partitions


6 Partitions


10 Partitions


Notice that the trapezoidal approximation leaves less gaps of space inside the curve, which means that the approximation is more accurate. In both cases, rectangular and trapezoidal, the more partitions we use, the more accurate our approximation is.

Formula: Trapezoidal Rule

In this article, we will estimate the definite integral using two methods:

  1. Applying the formula
  2. Using the area formula for trapezoids

Let’s apply the formula for the following example:

Example 1: Estimate the area under the curve of y = x2 on the interval of [0,2] using the trapezoidal rule.

  • Approximate the trapezoidal sum using 2, 4, and 6 trapezoids.
  • Use an integral calculator to find the exact area under the curve.
  • Compare your estimate of the area to the actual area under the curve.

Solution:

Let’s graph function y = x2 on the coordinate plane:

2 Trapezoids

First, we need to find the length of each sub-partition, or base, of each rectangle.

Use the formula ,

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n =2, so let’s apply the formula:

Therefore, the length of the base of each rectangle is 1.

Second, we will find the value of each x value going by 1 starting from a = 0:

x a = 0 1 b = 2
F(x) 0 1 4

From the table, f(x0)= 0, f(x1)= 1, and f(x2)= 4.

Third, we will apply the formula.

Let’s apply the formula:

4 Trapezoids

First, we need to find the length of each sub-partition, or base, of each rectangle.

Use the formula ,

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n = 4, so let’s apply the formula:

Therefore, the length of the base of each rectangle is ½.

Second, we will find the value of each x value going by ½ starting from a = 0:

x a = 0 1/2 1 1.5 b = 2
F(x) 0 0.25 1 2.25 4

From the table, f(x0)= 0, f(x1)= 0.25, f(x2)= 1, f(x3)= 2.25, and f(x4)= 4

Let’s apply the formula:

6 Trapezoids

First, we need to find the length of each sub-partition, or base, of each rectangle.

Use the formula

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n= 6, so let’s apply the formula:

Therefore, the length of the base of each rectangle is 1/3.

Second, we will find the y value of each x value going by 1/3 starting from a = 0:

x a = 0 1 4/3 5/3 b = 2
F(x) 0 0.111 0.444 1 1.778 2.778 4

From our estimates, T2 =3 square unitsT4=2.75 square units, and T6=2.70 square units. Let’s compare this value to the actual area under the curve using an integral calculator:

Our trapezoid estimates are overestimates of the curve. However, as mentioned earlier, the more trapezoids we use, the more accurate our estimates become.

Now we are going to apply the trapezoidal rule using the area formula of a trapezoid.

Example 2: Estimate the area under the curve of y = x2 on the interval of [0,2] using the trapezoidal rule.

  • Approximate the trapezoidal sum using 2, 4, and 6 trapezoids. Draw a diagram of your results.
  • Use an integral calculator to find the exact area under the curve.
  • Compare your estimate of the area to the actual area under the curve. Are your estimates over or underestimates?

Solution:

Let’s graph function y = x2 on the coordinate plane:

2 Trapezoids

First, we need to find the length of each sub-partition of each trapezoid.

Use the formula

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n =2, so let’s apply the formula:

Therefore, the height of each trapezoid is 1.

Second, we will find the value of each x value going by 1 starting from a = 0:

x a = 0 1 b = 2
F(x) 0 1 4

Third, we need to find the 2 side lengths of each trapezoid (b1 and b2). Our first trapezoid is on the interval of [0,1]. The y values at [0,1] will be the side lengths b1= 0 and b2= 1.

The area of the first trapezoid is therefore A1= 1/2*h *(b1+b2) = 1/2*1 *(0+1) = 1/2square units

Our second rectangle is on the interval of [1,2]. Our second trapezoid is on the interval of [1,2]. b1= 1 and b2= 4.

The area of the second rectangle is A2= 1/2*h *(b1+b2) = 1/2*1 *(1+4) = 5/2 square units

Fourth, let’s add the areas of each trapezoid:

T2=A1+ A2= 5/2 + 1/2 ≈ 3

The approximate area under the curve is 3 square units. Below is a diagram of the approximation:

Looking at the graph, this is an overestimate of the area because the trapezoids rise above the curve.

4 trapezoids

First, we need to find the length of each sub-partition of each trapezoid.

Use the formula

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n =2, so let’s apply the formula:

Therefore, the height of each trapezoid is ½.

Second, we will find the value of each x value going by ½ starting from a = 0:

x a = 0 1/2 1 1.5 b = 2
F(x) 0 0.25 1 2.25 4

Third, we need to find the 2 side lengths of each trapezoid (b1 and b2). Our first trapezoid is on the interval of [0,½]. The y values at [0,½] will be the side lengths. b1= 0 and b2= 0.25.

Fourth, let’s add the areas of each trapezoid:

T4=A1+ A2+ A3+A4= 0.0625+0.3125+0.8125+1.5625 ≈ 2.75

The approximate area under the curve is 2.75 square units. Below is a diagram of the approximation:

Looking at the graph, this is an overestimate of the area because the trapezoids rise above the curve.

6 Trapezoids

First, we need to find the length of each sub-partition of each trapezoid.

Use the formula

n = number of trapezoids

a = start of the interval

b = end of the interval

In the problem b = 2, a= 0, and n =6, so let’s apply the formula:

Therefore, the height of each trapezoid is ⅓.

Second, we will find the value of each x value going by ⅓ and starting from a = 0:

x a = 0 1 4/3 5/3 b = 2
F(x) 0 0.111 0.444 1 1.778 2.778 4

Third, we need to find the 2 side lengths of each trapezoid (b1and b2).

Fourth, let’s add the areas of each trapezoid:

T6≈A1+ A2+ A3+A4+ A5+A6= 0.0185+0.0925+0.2406+0.463+0.7593+0.7963 =2.7037

The approximate area under the curve is 2.7037 square units. Below is a diagram of the approximation:

Looking at the graph, this is an overestimate of the area because the trapezoids rise above the curve.

Using an integral calculator,

Summary of Section

The Trapezoidal Rule is a numerical method for evaluating the area between a curve and an axis by approximating the area with the areas of trapezoids.

The trapezoidal rule for approximating definite integrals is:

The trapezoidal rule can be utilized by two methods:

Using the formula:

  1. Determine the length of each sub-partition.
  2. Create a data table of values based on the increment determined by the length of each sub-partition.
  3. Apply the formula.

Using the area formula of trapezoids:

  1. Determine the length of each sub-partition.
  2. Create a data table of values based on the increment determined by the length of each sub-partition.
  3. Determine the dimensions of each trapezoid formed from the x values in the data table and the height (length of each sub-partition) of each trapezoid.
  4. Add the area of each trapezoid.

References:
https://www.math24.net/trapezoidal-rule/
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/understanding-the-trapezoid-rule

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