Since the definite integral is so closely tied to sums, you need to understand the basics of sigma notation and sums, found on this other page.
If you want a complete lecture on this topic, we recommend this video.
video by Prof Leonard |
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Summation and the Definite Integral
The general sum \(\displaystyle{ \sum_{i=1}^{n}{[f(x_i) \cdot \Delta x_i]} }\) is an approximation to the area under the curve \(f(x)\). Let's look more closely at this sum and get an idea of what is going on.
For each \(i\), we calculate the area of a rectangle on the interval \([a,b]\). The height of the rectangle is \(f(x_i)\) and the width is \(\Delta x_i\). If the same area is broken into more and more rectangles, we can get a better and better approximation of the area.
To get the actual area, we can take the limit of the above sum as follows.
\[
\lim_{n \to \infty}{ \left[ \sum_{i=1}^{n}{(f(x_i) \cdot \Delta x_i)} \right] } = \int_{a}^{b}{f(x)~dx}
\]
Let's break down the equation. When taking the limit we get
\(\displaystyle{ \lim_{n \to \infty}\sum_{i=1}^{n}{} ~~~ \to ~~~ \int_{a}^{b}{} }\) |
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\(\Delta x_i ~~~ \to ~~~ dx \) |
The idea is that the width of each interval goes to \(dx\). The limit of the sum goes to \(\int_{a}^{b}{}\). You can think of the integrand, \(f(x)\), being swept from \(a\) to \(b\).
Okay, time for a video. Here is a good video on the definition of the definite integral. It will give you an intuitive understanding what it means and how the notation works.
video by PatrickJMT |
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Before we actually evaluate the Reimann sum with limits, let's see how to approximate the value of a definite integral. This is a great video with explanation of an example.
video by PatrickJMT |
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From the first video in this section, you know that the limit given above is the definition of the definite integral using Reimann Sums. Here is an example showing how to calculate a definite integral using this definition. The example is worked in two consecutive videos.
video by PatrickJMT |
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video by PatrickJMT |
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Fortunately, we are not going to have to calculate these sums and limits all the time. The next page discusses the (First) Fundamental Theorem of Calculus, which we use extensively in calculus.
You CAN Ace Calculus
Definite integrals are usually introduced early in the study of integration after covering the basics and integration by substitution. However, some practice problems on this page require the use of integration by parts, which is a more advanced technique usually introduced in second semester calculus. If you haven't studied integration by parts yet, no worries. You can skip those practice problems and come back later, once you have covered integration by parts. |
external links you may find helpful |
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Wikipedia: Fundamental Theorem of Calculus (It is interesting that in wikipedia, the first and second fundamental theorems are switched, which is different than is sometimes taught in first semester calculus and discussed in Larson Calculus.) |
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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