9.6.2 An Example of the Trapezoidal Rule

Question 1

An Example of the Trapezoidal Rule

Answer 1

• The trapezoidal rule approximates the area A of the region bound by the curve of a continuous function f (x) and the x-axis using N partitions on [a, b].

Question 2

note

Answer 2

• Here’s a strange situation. Suppose you are stranded on a desert island with a group of people, and for some reason you need to know the value of the natural log of three. Hey, weirder things have happened!
• Well, if you remember from Calculus I that the integral of 1/x is equal to the natural log function, you can construct an integral whose area is equal to exactly ln 3.
• How could you use this information to determine the value of ln 3?
• Since the only tools you have to compute numbers are what you know, then you will have to approximate the value of ln 3.
• Here is the trapezoidal rule. If you use this rule, your
  approximation will be better than those of the other people stranded on the island.
• Here, the interval [a, b] is divided into 4 regions.
  So f (x) = 1/x, N = 4, a = 1, and b = 3.
• If you wanted an even more precise approximation, you could increase the number of rectangles used in the approximation.
• Each partition has a base of 1/2. Start by finding what the x-values are at the partitions. Then find their corresponding f (x)-values. Plug these values into the trapezoidal rule.
• Find a common denominator and add the fractions together.
• Notice that your answer is very close to the actual value of ln 3.
• Try doing this example with eight partitions to see if the approximation gets better. Also, compare the trapezoidal rule to the rectangular approximation.
• Here is another example. This time, all that is given is a table of values. The function itself isn’t known, but the integral can still be approximated using the Trapezoidal Rule.
• There are eight partitions and the endpoints are a = 2 and b = 5.2. The approximate value of the integral is 20.94, and this was obtained by using only the points in the table that lie on the curve.

Question 3

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫2 0 1/1+x^2dx.

Answer 3

A≈287/260

Question 4

Use the trapezoidal rule with 4 trapezoids of equal base to approximate ∫ 2 1 1/x^2dx.

Answer 4

A≈1/8[1+2(4/5)^2+2(2/3)^2+2(4/7)^`2+1/4]

Question 5

Use the trapezoidal rule with 4 trapezoids to find an expression that approximates ∫4 2 e^x dx.

Answer 5

1/4(e^2+2e^5/2+2e^3+2e^7/2+e^4)

Question 6

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 2 1 √x^2−1 dx.

Answer 6

A≈8+3√3+2√7/18

Question 7

Use the Trapezoidal Rule and the following data to estimate the value of the integral ∫ 2.2 1 y dx.

Answer 7

8.36

Question 8

Use the Trapezoidal Rule and the following data to estimate the value of the integral ∫ 2.8 0 y dx.

Answer 8

18.88

Question 9

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 3 2 1/1−x dx.

Answer 9

A≈−7/10

Question 10

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫ 2 0 x√x^2+4 dx.

Answer 10

A≈1/8[15+8√2+4√5+√17]

Question 11

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫π 0 sinx dx.

Answer 11

A≈π(1+√2)/4

Question 12

Use the trapezoidal rule with four trapezoids to find an expression that approximates ∫ 3 1 √x dx.

Answer 12

1/4[1+2√3/2+2√2+2√5/2+√3]

Question 13

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫1 0 x^3/2dx.

Answer 13

A≈5+2√2+3√3/32

Question 14

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 1 0 1−x/1+x dx.

Answer 14

A≈2/5