(a) Estimate the area under the graph of f(x)=√(x) from x=0 to x=4 using four approximating rect

(a) Estimate the area under the graph of f(x)=√(x) from x=0...

Key Concepts

Underestimate vs Overestimate in Riemann Sums

The terms underestimate and overestimate refer to whether a Riemann sum gives a value that is less than or greater than the true area under a curve. For example, when a function is increasing, using the left endpoints tends to underestimate the area, while using the right endpoints tends to overestimate it. Recognizing these tendencies is important when evaluating the accuracy and behavior of different numerical integration methods.

Integration as Area Under a Curve

Integration in the context of calculus is often interpreted as the accumulation of quantities, most notably the area under a curve within a given interval. This geometric interpretation links the concept of the definite integral to problems of finding areas, and Riemann sums provide a foundational approach to approximating these areas before taking a limit to arrive at the exact result.

Riemann Sum

A Riemann sum is a method for approximating the area under a curve by dividing the domain into subintervals, computing the function's value at certain sample points within these subintervals, and summing the products of these values with the subinterval widths. This technique forms the basis for numerical integration and understanding the definite integral as a limit of such sums.

Endpoint Approximation

Endpoint approximation refers to the method of selecting sample points at the endpoints of each subinterval when constructing a Riemann sum. There are two common types: using the right endpoint, where each sample point is chosen at the right edge of a subinterval, and using the left endpoint, where it is chosen at the left edge. The choice of endpoint can affect whether the approximation is an overestimate or an underestimate, particularly for monotonic functions.

Transcript

00:06 Problem we have a graph of f of x equals the square root of x so that looks like we'll start with part a part a we are using right end point rectangles to approximate the area between x equals zero and x equals four so we have here here is zero and here is four.

01:03 You are finding this area.

01:14 So for a, we want to use four rectangles.

01:35 And then we want to use our right end point.

01:39 That would be here.

01:44 So let's draw four rectangles, making blue.

01:52 And i'm just going to mark all the points that we have to use.

01:54 So with our right endpoints, it have to be used.

02:28 So the base of each rectangle is one.

02:34 And if our area equals the base times height, and we have to add up the base times height of each of these.

02:46 We have the base times height 1 plus the base times height 2 plus the base times height 3 plus the base times height 4.

03:05 These are all the same because the bases for each of these are all the same.

03:12 They all equal 1.

03:19 So what we can do is say that the base is equal to h1 plus, sorry, the area is equal to base times h1 plus h2 plus h3 plus h4.

03:32 And these points are here.

03:44 So then what we have to do is since b equals 1, that we can just ignore.

03:54 Because it's basically just like it's not even there.

04:00 So that means our area is equal to h1 plus h2 plus h3 plus h4, which is equal to f of 1 plus f of 2 plus f of 3 plus f of 4.

04:24 So when we plug these into our f of x equation, we plug these into our equation as x.

04:39 So we get a square root of 1 plus the square root of 2 plus the square root of 3 plus the square root of 4.

04:49 That the square root of 4 is 2, square root of 1 is just 1.

04:58 So we end up with 3 plus the square root of 2 plus the square root of 3...

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