MA005: Calculus I

5a. Use sigma notation to represent sums

• What is the purpose of using sigma notation?
• What is an example of a summation with a lower limit of 1 and an upper limit of 5 using j as the index?
• Define counter or index variable.
• What are some properties of summations?
• How would you write 1²+2³ +3⁴ + 4⁵ + ... +8⁹ in sigma notation with k as the index?

Sigma notation, or summation notation, is a shorthand way of writing the sum of many terms in a shortened form. The bottom limit (under the sigma) states which value the index begins with and declares which variable is the index. The top limit (above the sigma) tells you what is the last index value to use in your sum. The expression that follows the sigma (the summand) expresses the pattern to be followed as values are replaced with different index values. See Figure 5a.1.

Figure 5a.1.

To review, see Sigma Notation and Riemann Sums.

5b. Approximate the areas of regions

• What is the general idea behind approximating the area under a curve?
• When is the process exact?
• What happens to the accuracy as the number of rectangles increases?
• How does the right-hand rule differ from the left-hand rule?
• What is the procedure, in words, for the midpoint rule?

Areas of complex regions can be approximated by dividing up the region using shapes we already have geometric formulas for. In calculus, we generally use rectangles for a first approximation to estimate the area of the region. Divide up the x-axis into evenly sized segments $\Delta x$, and using the height of the function $\( f(x_i) \)$ as the second dimension. Selecting the right side, the left side, or the middle of the region provides a slightly different estimate, but as the number of rectangles used increases, so does the accuracy of the estimate.

To review, see Introduction to Integration.

5c. Approximate areas by Riemann sums

• How would you evenly partition the interval [3,7] for n=6?
• What is the upper sum estimate for the area under the curve f(x)=1/x on the interval [3,7] using six rectangles?
• What is the lower sum estimate for the area under the curve f(x)=1/x on the interval [3,7] using six rectangles?
• What is the procedure for finding a Riemann sum for general n? Describe the steps?

Riemann sums are used to estimate the area under a curve by cutting up the region into small rectangles, a process that becomes more accurate with more rectangles. The upper sum for a region is an overestimate of the area, and the lower sum for a region is an underestimate.

As the number of rectangles gets larger, both calculations for area will converge to the same value, the true area. Procedurally, divide up the interval over which the area will be calculated into a partition with n subintervals. 

If you are calculating an upper sum, determine at which endpoint in the partition (right or left on each interval) will produce the larger area, and then evaluate the function curve at that point. Multiply the height of the function at each partition's upper value by the width of each partition. Then add up the areas to obtain the estimate. See Figure 5c.1.

Figure 5c.1.

To review, see Sigma Notation and Riemann Sums.

5d. Translate an area under a curve into a definite integral

• What is the definite integral version of the shaded area in the figure? See Figure 5c.2.
• How do definite integrals relate to Riemann sums? Give at least two ways.
• What does it mean graphically if the area of a definite integral turns out to be negative?
• How can we interpret definite integrals with some area positive and some negative in terms of total net change?
• Define the integrand function.

Definite integrals are a way of representing area (and other concepts) instead of using Riemann sums. In general, the interval becomes the limits of integration on the integral symbol, with the smaller value on the bottom, and the larger one on the top.

The function that defines the height of the area goes after the integral sign. The expression is closed out by dx. See Figure 5d.1. When the function is above the axis, the area calculated will be positive. When the function is below the axis, the area calculated will be negative. 

Figure 5d.1.

Figure 5d.2.

To find the geometric area, all components will need to be positive before adding. To find the total net change, add up the signed values. See Figure 5d.3.

Figure 5d.3.

To review, see The Definite Integral.

5e. Evaluate definite integrals geometrically using graphs of functions

• What is the definite integral that represents the area under the curve shown in the figure? See Figure 5e.1.
• How could we use geometry to find the area represented by the integral?
• What are some geometric formulas we could apply to calculating areas? Draw an example of each.

Because one interpretation of integrals is that of area, we can use geometry to calculate integrals. For instance, a straight-line function might produce an area of a rectangle (if it is horizontal), a triangle (if it goes through zero on one endpoint), or a trapezoid. Or, we can combine several of these to find the area under a piecewise function. See Figure 5e.2.

Figure 5e.1.

Figure 5e.2.

To review, see Areas, Integrals, and Antiderivatives and Properties of the Definite Integral.

5f. Determine whether a given function is integrable

• What kinds of functions are integrable?
• Give an example of a function that is not continuous but is integrable.

More functions are integrable than are differentiable. To be integrable, a function need only be piecewise continuous (with a finite number of pieces) and bounded (no part of it goes to infinity on the interval we are interested in). Using properties of integrals, we can break out each piece on which the function is continuous, and compute the integral separately on each piece, and then add the pieces back up. See Figure 5f.1.

Figure 5f.1.

To review, see Properties of the Definite Integral, Areas, Integrals, and Antiderivatives, and Finding Antiderivatives.

5g. Use antiderivatives to evaluate a function

• What is the relationship between derivative rules and antiderivative rules?
• What is the power rule for antiderivatives?
• What is the exception to the power rule (for antiderivatives)?
• What are the common trigonometric antiderivatives?
• What is the antiderivative rule for exponential functions?
• Why do we add a constant of integration to indefinite integrals?

If we want to evaluate $\int x^2 dx$, we apply the power rule for antiderivatives, which says that $\int x^n dx = \frac{x^{n+1}}{n+1}+C$. In this situation, $n=2$, and following the rule, we end up with $\frac{x^3}{3}+C$. The constant of integration is required because the derivative of any constant is zero, and so this is information we are unable to recover completely. If you took the derivative of $\frac{x^3}{3}+1$ or $\frac{x^3}{3}-6$ or $\frac{x^3}{3}$, they would all produce the same function, $x^2$.

Some common antiderivative rules:

• $\int x^n dx = \frac{x^{n+1}}{n+1}+C$ for $x\neq -1$.
• $\int \frac{1}{x} dx = ln|x| +C$
• $\int \sin x dx = -\cos x +C$
• $\int \cos x dx = \sin x +C$
• $\int \sec^2 x dx = \tan x +C$
• $\int \sec x \tan x dx = \sec x +C$
• $\int e^x dx = e^x +C$

For more expressions with more terms, separated by plus or minus signs, integrate term by term.

To review, see Areas, Integrals, and Antiderivatives and Finding Antiderivatives.

5h. Find antiderivatives by changing the variable and using tables

• What is an example of an integral that requires substitution to integrate?
• What is an example of an integral that uses change of variables, but where the inside function rule-of-thumb will not work?
• How can substitution be used to assist us when using tables of integrals?

Many functions are built up from simple functions into more complicated functions. The change of variable technique permits us to simplify the functions again so that basic antiderivative rules can be applied. They also permit us to look up the general form of an integral in a table of integrals and match up the function we have with the form in the table.

To review, see Areas, Integrals, and Antiderivatives and Using Tables to Find Antiderivatives.

5i. Use the Fundamental Theorem of Calculus to evaluate definite integrals

• What is the Fundamental Theorem of Calculus and why is it so important?

The Fundamental Theorem of Calculus allows us to relate the value of an integral to the antiderivative of the function being evaluated at the limits of integration. 

To review, see Areas, Integrals, and Antiderivatives and The Fundamental Theorem of Calculus.

5j. Differentiate integrals

• What is the second part of the Fundamental Theorem of Calculus (Leibniz's Rule)?
• Explain how to apply the second part of the Fundamental Theorem to find the derivative for $f(x)=\int_0^{sin(x)}{2 t^2}$.

Area functions (or accumulation functions) and functions defined by integrals with variables in the limits. The second part of the Fundamental Theorem allows us to find derivatives for such functions, even when the limits of integration are functions themselves, rather than just a plain variable by incorporating the use of the chain rule.

To review, see The Fundamental Theorem of Calculus.

5k. Solve applied problems that involve generalized area, that is, distance, work, and so forth

• What are some examples of problems that can be solved by using generalized area? Express the integrals in terms of units rather than specific functions.

To calculate area, the function inside the integral represents the height of the function. By integrating, we obtain area (accounting for the second dimension). Similarly, if the function inside the integral represents area, then integrating finds the volume of the region. Since work is force times distance, and integral with a function for force produces work.

To review, see First Application of Definite Integrals.

5l. Find an area between two curves

• How do we find the area between two curves if there are no points where they cross?
• How does the technique above have to be modified if the curves do cross? What if they cross more than three times or more?

To find the area between two curves, first, it is necessary to determine which function is the larger (higher on the y-axis), and to determine if the functions cross at any point. The area between the curves is the difference between the two functions integrated over an interval (which may be specified or may be determined by points of intersection).

If the orientation of the functions changes (their positions switch at a cross point inside the interval), then the integral must be broken up into pieces using properties of integrals in order to change the orientation to obtain the geometric area. See Figure 5l.1.

Figure 5l.1.

To review, see First Application of Definite Integrals.

5m. Find the average (mean) value of a function

• What is a geometric interpretation of the mean (average) value of a function?
• What is the formula used to calculate the average value of a function?

We calculate the average value of a function by finding the area of the region, divided by the length of the interval between the limits of integration. This gives the average height of the function. It is also the height a rectangle would need to be in order to be equal to the area under the curve on the same interval. See Figure 5m.1.

Figure 5m.1.

To review, see First Application of Definite Integrals.

Unit 5 Vocabulary

• Accumulation function
• Antiderivative
• Area function
• Counter
• Definite integral
• Differential calculus
• Fundamental Theorem of Calculus
• Indefinite integral
• Index variable
• Integral calculus
• Integrand
• Integration
• Leibniz's Rule
• Limits of integration
• Limits of summation
• Lower sum
• Mean (average) Value Theorem (for integrals)
• Partition
• Riemann sum
• Sigma notation
• Summand
• Upper sum