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AP Calculus AB: 9.6.2 An Example of the Trapezoidal Rule

Mathematics14 CardsCreated 3 months ago

This section demonstrates how to apply the Trapezoidal Rule to approximate definite integrals, such as estimating ln f(x)=1/x over [1, 3]. It walks through partitioning the interval, calculating function values, and plugging them into the formula, emphasizing how increasing partitions improves accuracy—even when only discrete data points are available.

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An Example of the Trapezoidal Rule

• 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].

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Key Terms

Term
Definition

An Example of the Trapezoidal Rule

• 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...

note

  • 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 va...

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

A≈287/260

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

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

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

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

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

A≈8+3√3+2√7/18

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AP Calculus AB: 9.6.2 An Example of the Trapezoidal Rule
TermDefinition

An Example of the Trapezoidal Rule

• 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].

note

  • 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.

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

A≈287/260

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

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

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

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

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

A≈8+3√3+2√7/18

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

8.36

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

18.88

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

A≈−7/10

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

A≈1/8[15+8√2+4√5+√17]

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

A≈π(1+√2)/4

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

1/4[1+2√3/2+2√2+2√5/2+√3]

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

A≈5+2√2+3√3/32

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

A≈2/5