9.6.1 Deriving the Trapezoidal Rule

Question 1

Deriving 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

• When you take the integral to find the area under a curve, you are actually dividing the region into an infinite number of rectangles of infinitesimally small width and adding the areas of those rectangles together. But some functions can’t be integrated. In these cases, you can approximate the definite integral and the area under the curve with a finite number of rectangles.
• But notice that rectangles produce a lot of error.
• However, trapezoids produce less error. This is because the trapezoid more closely emulates the partition that it is approximating.
• To approximate an area using trapezoids, you first have to know how to find the area of a trapezoid. The area of a trapezoid is equal to the base multiplied by the average of its two heights.
• Suppose you have a curve bounded on the interval [a, b] and you need to approximate the area.
• Start by deciding how many partitions you want to find. The more partitions, the more accurate your approximation will be. By the same token, more partitions also means more calculations. The number of partitions is called N.
• Find the value of f for each x-value where you are breaking up your interval. Plug these values into this formula to approximate the area. This formula is called the trapezoidal rule.
• There are a couple of pieces of the trapezoidal rule to watch out for. First, notice that you will have to evaluate N + 1 terms if you use N partitions. Also notice that there are several two’s floating around the formula. Don’t leave any of those out!

Question 3

Which of the following is most accurate?

Answer 3

Trapezoids produce approximations equal to or better than those of rectangles.

Question 4

Estimate the area under the curve y = f (x) = x ^2 from x = 1 to x = 4 using three trapezoids with bases of equal length.

Answer 4

21.5

Question 5

What is the formula for the area of this trapezoid?

Answer 5

B(H1+H2/2)

Question 6

Use six trapezoids with bases of equal length to estimate the area under the curve f(x)=√9−x^2

from x = −3 to x = 3.

Answer 6

3+2√5+4√2

Question 7

True or false?

When approximating the area under a curve using trapezoids, the bases of the trapezoids must be the same widths.

Answer 7

false

Question 8

Use the trapezoidal rule to estimate the area under the curve g (x) = 3x + 5 from x = 0 to x = 3.

Answer 8

57/2

Question 9

Use the trapezoidal rule with 3 trapezoids to approximate the integral.
∫ 4 1 1/x dx

Answer 9

35/24

Question 10

You wish to use trapezoids to estimate the area under the curve y = x 3 from x = 1 to x = 4. For a quick estimate, what is the best number of trapezoids to use?

Answer 10

3

Question 11

Estimate the area under the curve y = f (x) = |x| + 2 from x = −2 to x = 2 using four trapezoids with bases of equal length.

Answer 11

12

Question 12

Estimate the area between the x-axis and the curve y=f(x)=cos x from x=0 to x=π using four trapezoids with bases of equal length.

Answer 12

π/4(1+√2)