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## Integral Calculus (2017 edition)

### Course: Integral Calculus (2017 edition)>Unit 1

Lesson 1: Definite integral as area

# Warmup: Definite integrals intro

Practice evaluating definite integrals by finding the area using shapes (like rectangles & circles) under a function.

## Problem 1

The graph of a function f is shown below.
a) integral, start subscript, minus, 6, end subscript, start superscript, 1, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals

b) integral, start subscript, 1, end subscript, start superscript, 4, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals

c) integral, start subscript, minus, 6, end subscript, start superscript, 4, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals

## Problem 2

The graph of a function g is shown below. The graph consists of a semicircle and linear segments.
a) integral, start subscript, minus, 7, end subscript, start superscript, minus, 1, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals

b) integral, start subscript, minus, 1, end subscript, start superscript, 8, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals

c) integral, start subscript, minus, 7, end subscript, start superscript, 8, end superscript, g, left parenthesis, x, right parenthesis, d, x, equals

## Challenge problem

The graph of a function h is shown below. The graph consists of a semicircle and linear segments.
a) integral, start subscript, minus, 7, end subscript, start superscript, minus, 2, end superscript, h, left parenthesis, x, right parenthesis, d, x, equals

b) integral, start subscript, minus, 5, end subscript, start superscript, 1, end superscript, h, left parenthesis, x, right parenthesis, d, x, equals

c) integral, start subscript, minus, 1, end subscript, start superscript, 8, end superscript, h, left parenthesis, x, right parenthesis, d, x, equals

## Want to join the conversation?

• The Definite Integrals Introduction section should be after the Riemann Sums sections. Otherwise, the notations used make no sense (I didn't encounter them until now). Is there a problem with the order of the sections or am I wrong with something?
• I can understand what many of you are thinking, so let’s examine what is really happening here.

In the section previous to this you were introduced to the notation and concept of the indefinite integral, and a few rules of integration of various functions in as much as to demonstrate integration being the inverse of differentiation.

In this section you are introduced to the notation of the definite integral, (which is essentially the same as the indefinite integral in the previous section, with the only difference being that the limits of integration are added) and you are told that the definite integral is going to be used to find the area under the curve. If you look carefully, however, in none of the questions/examples in this section are you actually being asked to integrate using the rules of integration. In all the examples, the area is determined using, by what should be by now well known to you, geometric formulas for area (circles, squares, triangles). So at this point the curious student should be thinking along the lines of, “well that was certainly trivial, I haven’t been shown anything I couldn’t do before without the integral notation, so tell me, what happens when the curves of the functions are not so easily described with just lines and semi -circles?” . . . .

In the next section you learn an intuitive “summing method to approximate area under a (relatively) arbitrary curve,” and, how via using a limiting process on the summing method the exact area can be found leading to why the notation came about.

This is a typical “show them where they’re going, then show them how to get there” approach; and you will see it a lot – and equally as often, once you have been shown how to get there, you still won’t have enough mathematical maturity to understand why it works. Depending on your reasons for studying calculus, for example, if you just want to graduate in another discipline that simply requires calculus, and not graduate in mathematics, you may never learn why a process works.

This teaching style might not be to your personal liking, but it is not at all uncommon.
• This section was confusing, and even though I've figured it out now, I still don't get why it's presented like this. Usually Sal introduces a concept, even if only the basic notation and function, before Khan asks you to try it yourself. This worksheet shoves the learner into unfamiliar territory, assuming (s)he knows exactly what the new notation means. Maybe it expects him/her to ask for the hint...? But the hint still assumes (s)he knows that a definite integral can be represented by the area between the function in the x-axis: "Until now, we've been using approximate area to estimate the definite integral." I've calculated that sort of area just once or twice before in differential calc., and I'm pretty sure the exercises didn't mention the area-integral relationship. It just doesn't make sense to put this worksheet here without a concept video/article.
• User Stefen answered user Datcu Octavian's question with an amazing and extremely helpful answer in this same Question list. Look for it until you find it as it is a (near) perfect answer to your question.
• This was really difficult and not particularly educational because there was very little explanation.
• It is meant to be an introductory approach to the visualisation/intuition of what integral calculus is....graphically speaking is the area under the curve...This is true for single integrals....(double integrals give you the volume of the shape you are studying!)
• I got confused because I could not find explanation for the connection btw area and integral.
But I found them on YouTube.
• Thank you very much, and I agree--this video (and its part 2) define why a definite integral defines area, why the Integral symbol is an elongated S (for "sum"). It was clearly cut from this chapter and no updated version made.

I don't know why so many people are giving a "take it or leave it", "it's by design" stance in answering other questions on this matter. In the first video of the next section ( https://www.khanacademy.org/math/ap-calculus-ab/definite-integrals-intro-ab/definite-integral-properties-ab/v/integrating-scaled-function ) at , Sal says "When you think back to the Riemann sums". It's clearly a screw-up or oversight in updating the site/chapter structure, and should be fixed.
• Although I found this section very straightforward (because I learned all this 35 years ago!), I agree with most of the comments below - if not a video then an introductory passage is required explaining that definite integrals can be used to determine net areas. Plus an explanation of the specific notation for definite intervals... Please amend accordingly!
• I don't understand why they assume we know what it means to evaluate a definite integral. The previous section focused on indefinite integrals and if we just followed that course, we would've only learned about finding the anti-derivatives of common functions. Are we somehow meant to realize that evaluating a definite integral means to find the area under a curve in a given interval? What I'm complaining about isn't the fact that they didn't teach us how to find definite integrals first before introducing this article, but the fact that they didn't explicitly say what it meant to evaluate a definite integral. If it wasn't for the title of the article, I probably wouldn't have found out that the definite integral had something to do with area.
• It seems something is missing here? I checked backwards and forwards but couldn't find any videos explaining why definite integral is related to area under the curve.