You probably remember Becca and her track meet from earlier lessons. She won her race after pulling away from the pack in a hard push at the finish, and her boyfriend got a great picture of her just as she began pulling away. We have learned that by using derivatives, she could actually calculate her speed at the moment the picture was taken, and then with the second derivative she could calculate her acceleration similarly.
In this lesson we will discuss the integral, which is the process that would allow Becca to calculate the distance she actually covered during a given interval of the race, even though her speed was not constant!
Area Under the Curve
To understand integration, consider the area under the curve y = f(x) for the interval from x = a to x = b in the figure below.
One way to calculate the area is to fill the region with rectangles. If the region is curved, the rectangles will not fit exactly, but we can improve the approximation by using rectangles of thinner width. If we continue to make the rectangles thinner and thinner, the area under the curve would reach the exact area under the curve. This is the limiting process that we discussed. In other words, the area under the curve is the limit of the total area of the rectangles as the widths of the rectangles approach zero.
Consider again the figure above. The interval from x = a to x = b is subdivided into n equal subintervals. Rectangles are drawn in each subinterval. Each rectangle touches the curve at its upper right corner. The height of the first rectangle is f(x_{1}), the second f(x_{2}), and the last is f(x_{n}). Since the length of the entire interval from a to b is b  a, then the width of each subinterval is \begin{align*}\frac{ba}{b}\end{align*}




 \begin{align*}\Delta x = \frac{b  a} {n}\end{align*}
Δx=b−an
 \begin{align*}\Delta x = \frac{b  a} {n}\end{align*}



is defined as the width of each subinterval. The area of the first rectangle is \begin{align*}f(x_1)\Delta x\end{align*}
\begin{align*}A_n\end{align*} 
\begin{align*}= f(x_1) \Delta x + f(x_2) \Delta x + . . . + f(x_n) \Delta x\end{align*} 


\begin{align*}= \sum_{i = 1}^n f(x_i) \Delta x\end{align*} 
To make use of the concept of limit, we make the width of each rectangle approach 0 , which is equivalent to making the number of rectangles, n, approach infinity. By doing so, we find the exact area under the curve,



 \begin{align*}\lim_{n \rightarrow \infty} A_n = \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} f(x_i) \Delta x\end{align*}.


This limit is defined as the definite integral and it is denoted by



 \begin{align*}\int_{a}^{b} f(x) dx\end{align*}.


The Definite Integral
A definite integral gives us the area between the xaxis and a curve over a defined interval.
The Definite Integral (The Limit Method)


It is important to keep in mind that the area under the curve can assume positive and negative values. It is more appropriate to call it “the net signed area”. Example 2 below illustrates this point.
Examples
Example 1
Calculate the area between the curve y = x^{2} and the xaxis from x = 0 to x = 1.
We divide the region into n number of subintervals, each of width ∆x (see figure below).
First find ∆x.
\begin{align*}\Delta x\end{align*}  \begin{align*}= \frac{b  a} {n}\end{align*}  

\begin{align*}= \frac{1  0} {n}\end{align*}  
\begin{align*}= \frac{1} {n}\end{align*} 
The next step is to find x_{i}.
\begin{align*}x_i\end{align*}  \begin{align*}= a + i \Delta x\end{align*}  

\begin{align*}= 0 + i \cdot \frac{1} {n}\end{align*}  
\begin{align*}= \frac{i} {n}\end{align*} 
Therefore, \begin{align*}f(x_i) = x^2_i = \left (\frac{i} {n}\right )^2\end{align*} Using the integration formula
\begin{align*}A\end{align*}  \begin{align*}= \int_{a}^{b} f(x) dx = \lim_{n \rightarrow \infty} \sum_{i = 1}^n f(x_i) \Delta x\end{align*}  

\begin{align*}= \int_{0}^{1} x^2 dx= \lim_{n \rightarrow \infty} \sum_{i = 1}^n \left (\frac{i} {n}\right )^2 \left (\frac{1} {n}\right )\end{align*}  
\begin{align*}= \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \frac{i^2} {n^3}\end{align*} 
Since we are summing over i, not n, the summation becomes,
\begin{align*}A\end{align*}  \begin{align*}= \lim_{n \rightarrow \infty} \frac{1} {n^3} \sum_{i = 1}^{n} i^2\end{align*}  

\begin{align*}= \lim_{n \rightarrow \infty} \frac{1} {n^3} (1^2 + 2^2 + 3^2 + . . . + n^2)\end{align*} 
But since \begin{align*}\sum_{i = 1}^{n} i^2 = \frac{n(n + 1) (2n + 1)} {6}\end{align*} then
\begin{align*}A\end{align*}  \begin{align*}= \lim_{n \rightarrow \infty} \frac{1} {n^3} \frac{n(n + 1)(2n + 1)} {6}\end{align*}  

\begin{align*}\lim_{n \rightarrow \infty} \frac{1} {6} \left (2 + \frac{3} {n} + \frac{1} {n^2}\right )\end{align*} 
Taking the limit,
\begin{align*}A\end{align*}  \begin{align*}= \frac{1} {6} (2 + 3(0) + (0))\end{align*}  

\begin{align*}= \frac{1} {3}\end{align*} 
Thus the area under the curve is (1/3).
Example 2
Find the area between the curve y = x and the xaxis from x = 1 to x = 1.
As you can see in figure a, the integral represents the total areas of all the rectangles above and below the xaxis. First, we divide the region into two regions, one above xaxis and one below the xaxis. Then we divide each region into n subintervals, each of width ∆x (figure b).
Region I: Find ∆x and x_{i}.
\begin{align*}\Delta x = \frac{1  0} {n} = \frac{1} {n}\end{align*}
\begin{align*}x_{i}\end{align*}  \begin{align*}= a + i \Delta x\end{align*}  

\begin{align*}= 0 + i \left (\frac{1} {n}\right ) = \frac{i} {n}\end{align*} 
\begin{align*}f(x_i) = \frac{i} {n}\end{align*}
Region II: Again, find ∆x and x_{i}.
\begin{align*}\Delta x\end{align*} \begin{align*}= \frac{1 0} {n} = \frac{1} {n}\end{align*}
\begin{align*}x_i\end{align*}  \begin{align*}= b + i \Delta x\end{align*}  

\begin{align*}= 1 + i \left (\frac{1} {n}\right )\end{align*}  
\begin{align*}= 1  \frac{i} {n}\end{align*} 
\begin{align*}f(x_i) = 1  \frac{i} {n}\end{align*}
The integral represents the net area of the two regions I and II:
\begin{align*}A = A_1  A_2 =\end{align*} \begin{align*}(\text{area above xaxis in }[a, b])  (\text{area below xaxis in }[a, b])\end{align*} Thus,
\begin{align*}A\end{align*}  \begin{align*}= \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} f(x_i) \Delta x  \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} f(x_i) \Delta x\end{align*}  

\begin{align*}= \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \left (\frac{i} {n}\right ) \left (\frac{1} {n}\right )  \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \left (1  \frac{i} {n}\right ) \left (\frac{1} {n}\right )\end{align*}  
\begin{align*}= \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \left (\frac{i} {n^2}\right )  \lim_{n \rightarrow \infty} \sum_{i = 1}^{n} \left (\frac{1} {n}  \frac{i} {n^2}\right )\end{align*}  
\begin{align*}\mathit = \lim_{n \rightarrow \infty} \frac{1} {n^2} \sum_{i = 1}^{n} i  \left [\lim_{n \rightarrow \infty} \frac{1} {n} + \lim_{n \rightarrow \infty} \frac{1} {n^2} \sum_{i = 1}^{n} i\right ]\end{align*}  
\begin{align*}= \lim_{n \rightarrow \infty} \frac{1} {n^2} \frac{n(n+1)} {2}  \left [0 + \lim_{n \rightarrow \infty} \frac{1} {n^2} \frac{n(n + 1)} {2}\right ]\end{align*}  
\begin{align*}= \frac{1} {2}  \left [\frac{1} {2}\right ]\end{align*}  
\begin{align*}= 0\end{align*} 
We conclude that the net area is zero.
Example 3
Approximate the definite integral between x = 0 and x = 40 by calculating the areas of rectangles which fill the area in the image below. Use at least 3 successively narrower sizes of rectangles.
The equation of the curve in the image is: \begin{align*}y = 60 40(1  \frac{9}{10}^x)\end{align*}.
\begin{align*}\therefore\end{align*} the closest approximated area is 1173.44 units (The actual calculated area is 1174.0373)
Example 4
Approximate the area under y = x + 3 on the interval [5,6] using the middle Riemann Sum with 5 subintervals.
Sketch of graph:
First, we divide the interval [5,6] into pieces:
Between x = 5 and x = 5.2, the middle value is 5.1 + 3 = 8.1
Between x = 5.2 and x = 5.4, the middle value is 5.3 + 3 = 8.3
Between x = 5.4 and x = 5.6, the middle value is 5.5 + 3 = 8.5
Between x = 5.6 and x = 5.8, the middle value is 5.7 + 3 = 8.7
Between x = 5.8 and x = 6, the middle value is 5.9 + 3 = 8.9
Adding these, we get 42.5.
To get the Riemann sum, take this answer and multiply by the width of each segment: 0.2
\begin{align*}\therefore 8.5\end{align*} is our approximated area.
Example 5
Approximate the area between y = 3x^{2} + x + 5 and the xaxis on the interval between x = 2 and x = 5 using the right Riemann Sum with 2 subintervals.
First, we divide the interval [2,5] into subintervals: Between x = 2 and x = 3.5, the right value is 3(3.5)^{2} + (2.5) + 5 = 42.25 Between x = 3.5 and x = 5, the right value is 3(5)^{2} + (5) + 5 = 85 Adding these, we get 127.25. Take this answer and multiply by the width of each segment: 1.5.
\begin{align*}\therefore \approx 190.88\end{align*} is the area
Example 6
Use a definite integral to find the area under the curve \begin{align*}y = 5x^2 + 2x + 4\end{align*} on the interval [0, 3].
\begin{align*}\int_{0}^{3} 5x^2 + 2x + 4dx = \frac{5}{3} x^3 + x^2 +4x _{4}^{5}\end{align*}
Review
 Use the limit method to find the area under the curve of f(x) = x^{2} in the interval [0, 2].
Find the area between the curve and the xaxis:
 Curve y = x on the interval x = 1 to x = 3.
 Curve y = x from x = 1 to x = 3.
 Curve y = x from x = 3 to x = 3.
 Approximate the area under \begin{align*}y = 2x + 3\end{align*} on the interval [0,3] using the middle Riemann Sum for y with 6 subintervals.
Find the area under the curve:
 \begin{align*}y = 3\end{align*} on [4, 5]
 \begin{align*}y = 3x + 1\end{align*} on [1, 5]
 \begin{align*}y = \frac{1}{x}\end{align*} on [3, 4]
 \begin{align*}y = 2x + 4\end{align*} on [5, 6]
 \begin{align*}y = 5x^3 + 4x^2 + x + 2\end{align*} on [2, 5]
 \begin{align*}y = \frac {1}{x} \end{align*} on [3, 7]
 \begin{align*}y = 3x^2 + 2x\end{align*} on [5, 6]
 \begin{align*}y = 4\end{align*} on [2, 6]
 \begin{align*}y = 2x^2 + 4x + 5\end{align*} on [1, 5]
 Sketch y = x^{2} and y = x on the same coordinate system and then find the area of the region enclosed between them.
Review (Answers)
To see the Review answers, open this PDF file and look for section 8.12.