With a programmable calculator (or a computer), it is...

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $ n $, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $ n $ = 10, 30, 50, and 100. Then guess the value of the exact area. The region under $ y = x^4 $ from 0 to 1

Transcript

00:03 So this problem, they give us a function y is equal to x to the fourth, and roughly you should know that looks like a very steep version of a parabola from 0 to 1.

00:15 And we're asked to estimate the area using rectangles.

00:21 They ask us to use n equal 10, 30, 50, and 100.

00:24 And fortunately, we are doing this with calculating tools instead of doing this manually.

00:29 So if n is equal to 10, then on the interval from 0 to 1, the width of each rectangle is 110.

00:37 And then likewise, if n is equal to 30, 50, and 100, that's where we're going to come up with the width of these triangles, each getting smaller and smaller, as we have more and more, not triangles, but rectangles to do our estimates for the area.

00:52 So now, let's do this.

00:54 If i'm using right endpoints, so if i'm using a right endpoint, so think about this on the interval from 0 to 1, the right endpoint, that first endpoint is going to happen based on this width.

01:07 So in the first case, that first endpoint is going to be 0 .1.

01:10 And the second case, when n is 30th, 30, that's going to be 1 .30th, and then 1 .50th and so forth.

01:17 Okay.

01:17 So, and then we'll just need to figure out that tells me the width, and then evaluate the function to get the height.

01:23 So we're going to do all of this on our calculator.

01:26 So what we're going to do first is let's store this function.

01:28 So x to the 4th.

01:30 I'm going to store that function and just call that f of x for that as function.

01:49 Oh, sorry about that.

01:50 I got x to the fourth and then move the cursor down and then store that as f of x.

01:59 And i'm done.

02:00 So now i can evaluate f of any value and come up with that.

02:04 So i've stored f.

02:05 Now i need to create a sequence to get those right endpoints.

02:08 So i'm going to index with i.

02:11 And where does that first endpoint start? when n is equal to 10 that first endpoint is at one -tenth so the first endpoint is going to be at 0 .1 so that's the first right in point and then i'm going to go how far how far am i going i'm going all the way over to one and i'm going to increment that by point one so that's going to generate all the right endpoints for the rectangles so i need to store this so i'm going to store this in the variable we'll call this all of our x values so that's going to generate.

02:44 That gives me all of the x values i need in the case, and it gives me 10 of those to make that work.

02:50 Now, so that gives me the right end point of the right side of the rectangles.

02:55 I know the width is 1 tenth.

02:57 So to get all of these, i'm going to have to sum up.

02:59 So i'm going to have to have the width of the triangle is 0 .1.

03:04 So 0 .1 times the sum.

03:08 I need to add up.

03:09 Well, i need to evaluate f at each of those points...