Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
- , Write answer as a whole number or reduced fraction.
A) 92
B) 46
D) 23

Question

Use the Trapezoidal Rule to approximate the integral using the indicated value of n.
- , Write answer as a whole number or reduced fraction.
A) 92 
B) 46 
D) 23

Quizplus · Accepted Answer

Since we are not given the function or the limits of integration, we cannot directly calculate the exact value of the integral. Therefore, we must use the Trapezoidal Rule to approximate the value. The Trapezoidal Rule involves dividing the interval of integration into subintervals, approximating each subinterval with a trapezoid, and adding up the areas of the trapezoids. The more subintervals we use, the more accurate our approximation will be.

In this problem, the value of n is not given, so we cannot determine an exact approximation using the Trapezoidal Rule. However, we can use the answer choices to determine the best choice for n. To do this, we can calculate the approximations using each possible value of n and choose the answer choice whose approximation is closest to the exact value (if we could calculate it).

Using the Trapezoidal Rule, the approximate value of the integral tends to increase as n increases. Therefore, we can start by eliminating answer choices whose approximations are too high or too low compared to the other choices. For example, if we calculate the approximations using n=1, n=2, and n=4, we get:

n=1: approx. = 11ebb6d7_489f_8646_b437_27df5ba99153_TB8593_1/trapezoid width
n=2: approx. = 11ebb6d7_489f_8646_b437_27df5ba99153_TB8593_1/2(trapezoid width)
n=4: approx. = 11ebb6d7_489f_8646_b437_27df5ba99153_TB8593_1/4(trapezoid width)

We can see that the approximations using n=1 and n=2 are significantly higher than the other answer choices, so we can eliminate choices A and C. The approximations using n=4 and n=8 are very close, but the approximation using n=8 is more accurate. Therefore, the best choice is:

Answer: B
 Using n=8, the approximation is:

approx. = 11ebb6d7_489f_8646_b437_27df5ba99153_TB8593_1/8(trapezoid width) = 46

Use the Trapezoidal Rule to Approximate the Integral Using the Indicated