3. Explain geometrically how the Trapezoid Rule is used to app...

Question

3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says which of the following best describes the geometric process of using the trapezoid rule to approximate the interval from 2 to 8 of GFX DX. And we're given 4 possible choices as our answers. For choice A, we have divide the closed interval from 2 to 8 into N equal. Intervals draw rectangles under GFX and some their areas. For choice B, we have divided the closed interval from 2 to 8 into N equal subintervals, connect adjacent points on GFX with straight lines to form trapezoids, and some of the areas. For choice C, we have divide the close interval from 2 to 8 into N equal subintervals. Use the midpoint of each to draw straight lines to form squares, and sum the area under these lines. And for choice D, we have divided the close interval from 2 to 8 into N equal subintervals, use left end points to form rectangles, and some their areas. So we're asked to describe the geometric process of using the trapezoid rule to approximate our integral. So the first step in our process is to divide our interval into N equal subintervals, and the interval that we're going to be looking at is going to be the close interval from 2 to 8. Since our limits of integration go from X equals 2 to X equal to 8. For the second step, we want to form trapezoids on each of our sub-intervals since we're looking at the trapezoid rule. Now, the height of our trapezoid is just going to be the width of our subinterval. So that means that our height H is just going to be equal to the quantity of B minus A and quantity divided by N, where B here is the upper bound of our interval, A is the lower bound, and N is the number of some intervals. Now, the parallel sides of each trapezoids are going to be the function values at the endpoints of the sub-intervals. So, to form trapezoids, we just need to connect adjacent end points on GFX with straight lines. And then the final step is to sum up the area of all of our trapezoids. In doing so, we'll actually get our formula for the trapezoid rule, which is going to be H divided by 2. Multiplied by the quantity. Of G of That's not. Plus 2 multiplied by G of X1. Plus 2 multiplied by G of X2. Plus, and this pattern continues up until we get plus 2 multiplied by G of X N minus 1. Plus G of X of N. In quantity So, to summarize the process, we divide our interval into N equal sub-intervals. We connect adjacent points on GFX with straight lines to form trapezoids, and then some their areas. And so if we look at our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

Key Concepts

Definite Integral

Trapezoid Rule

Geometric Interpretation

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