Use finite approximations to estimate the area under the graph of the function using a. a lower

Use finite approximations to estimate the area under the...

Key Concepts

Upper Sum

The upper sum is another type of Riemann sum that uses the maximum function value within each subinterval to set the rectangle heights. This approach generally provides an overestimate for the area under the curve for increasing functions.

Equal Subdivision Partitioning

Partitioning the interval of integration into equal subintervals is essential for simplifying computations in Riemann sums. Using equal-width subintervals standardizes the calculation of rectangle areas and facilitates systematic refinement of the approximation by increasing the number of rectangles.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve by dividing the area into smaller shapes, usually rectangles, and summing their areas. This approach is foundational in integral calculus as it provides a way to approximate definite integrals, which represent the area under a curve.

Lower Sum

The lower sum is a specific type of Riemann sum where the height of each rectangle is determined by the minimum function value within each of the subintervals. This method tends to underestimate the actual area under the curve when the function is increasing over an interval.

Numerical Integration

Numerical integration encompasses a variety of techniques, including Riemann sums, for approximating the value of definite integrals when an antiderivative is difficult or impossible to find analytically. This field provides practical tools for estimating areas under curves and solving integral equations in applied contexts.

Transcript

00:01 And problem number one, we're asked to use finite approximations to estimate the area under the graph of f of x equals x squared from x equals 0 to 1.

00:10 Using in part a, we're going to use a lower sum with two rectangles of equal width.

00:16 So if we divide this 0 to 1 and two suburbables, we're going to get 0 .0 .5 and 0 .5 to 1.

00:24 And for a lower sum we want to take a rectangle and we want to use the lowest value of f of x on that sub interval for the height of our rectangle so from zero to zero point five our lowest value is actually zero so our first rectangle has no height on 0 .5 to 1 our lowest one is here at x equals 0 .5 so this is our second rectangle we can see that this is really not going to be a good approximation so for part a, we get, now we just sum the areas of these rectangles.

01:01 So i say that the area, sorry, the area is going to equal zero times, that's height, times width of 0 .5.

01:14 That's just going to be 0 plus height of 0 .5 squared, which is 0 .25 times the width of 0 .5.

01:24 So overall our area for part a we get 0 .125 in part b we're asked to do the lower sum of four rectangles so now we're gonna have four sub intervals and we'll get the same thing here each one's gonna be point two five wide we've got a width here the delta x of 0 .2 0 .25 so our first rectangle is it's a lower some is again going to have zero height second one will look like this second one look like this and the fourth one would look like this and so we have this should be 0 .25 .5 .5 .5 .5 and 1.

02:20 So for part b, go in box that in part b our area and again just add them up we get 0 .25 plus 0 .25 squared which is 0 .0625 times 0 .25 plus 0 .5 f of 0 .5 is 0 .25 times the delta x is 0 .25 plus 0 .75 squared is 0 .5625 times 0 .25 and so our area with four rectangles and of equal width and a lower sum.

03:11 It's going to be 0 .0625 times .25.

03:16 Let's factor that out.

03:18 Just factor that out.

03:19 You can do 0 .25, which is delta x times 0 .0625 plus 0 .25 plus 0 .56 plus 0 .25 plus 0 .5625 plus 0 .5 plus 0 .5625 plus 0 .5 plus 0 .5 plus 0 .5 plus 0 .5 plus 0625 plus.

03:48 15625, all times .25 is the area is going to be 0 .21875.

04:01 And so let's get in a little bit better approximation of area, but it's still a under approximation because we're using the lower sum.

04:10 Part c, we're asked to use a upper sum home.

04:16 Oh, i'm sorry.

04:18 I thought my recording had not been going this whole time.

04:22 At a minor panic attack, my apologies.

04:26 Part c, we're asked to have an upper sum with two rectangles of equal width.

04:31 So we're still going to have two, and our delta x is going to be 0 .25, delta x.

04:40 And so our upper sum, we take the highest point on the sub interval to be the height of our rectangle.

04:49 So from 0 .5, our highest point is here, and that's our rectangle.

04:54 And at 1, ice point is here, and that's our rectangle...

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