There are many people who deserve recognition for the important role they played in the development of this text. In the following chapters we are going to think in "the other direction." That is, given a func onf(x), we are going to consider our funcF(x) such thatF′(x) = f(x).
An deriva ves and Indefinite Integra on
The function in which we want to find a derivative is called the integrand. Now we can go "the other way": the derivative of an acceleration in the function gives a function of the velocity, etc.
The Definite Integral
To answer questions about the height of the object, we need to find the object's position function(s). Find the maximum speed of the object and its maximum displacement from its star position.
Riemann Sums
S Using the formula derived earlier, using 16 equal intervals and the right-hand rule, we can approximate the definite integral as The exact value of the definite integral can be calculated using the limit of a Riemann sum.
The Fundamental Theorem of Calculus
In (b), the height of the rectangle is less than fon[1,4], therefore the area of this rectangle is less than ∫4. The definite integral can be used to find the "area under a curve." List two other uses for definite integrals.
Numerical Integra on
Example 154 Integrating by Substitution: Inverse Trigonometric Functions Evaluate the given indefinite integrals. In his heart (using the remark of Theorem 44) subs tu he transforms integrals of the form∫. As substitutions "nullify" the chain rule, integration by parts "nullifies" the product rule.
In Exercises 15 – 23, use Subs tu on to evaluate the indefinite integral involving trigonometric functions. In Exercises 31 – 34, use Subs tu on to evaluate the indefinite integral involving logarithmic functions. In Exercises 41 – 50, use Subs tu on to evaluate the indefinite integral involving inverse trigonometric functions.
Integra on by Parts
This means that the integral on the right side of the formula Integra on by Parts,∫. Putting all of this together in the Integra on by Parts formula makes things work. Of course, we can use Integra on by Parts to also evaluate definite integrals, such as theorems.
In general, Integra on by Parts is useful for integrating certain products of functions, such as∫. T/A: Integration by Parts is useful in evaluating integrands containing products of functions. T/F: Integration by Parts can be considered the "opposite of the Chain Rule".
Trigonometric Integrals
S The powers of both the sine and cosine terms are odd, so we can apply the techniques of Key Idea 11 to any possibility. S The powers of sine and cosine are both equal, so we use the power-reducing formulas and algebra as follows. The cos3(2x)term is a cosine function with an odd power, which requires a subs tu on as done before.
If, for example, the sine power was odd, we extracted a sinx and converted the remaining even power of the sinx to a function using powers of cosx, leading to easy subduction. Integrate the first integral with subs tu on,u=tanx; integrate the second using rule #4 again. The next section introduces an integrated technique known as Trigonometric Substu, a clever combination of Subs and the Pythagorean Theorem.
Trigonometric Subs tu on
S We start by completing the square, then do subs tu- onu=x+3, followed by trigonometric subs tu on ofu=tanθ:. Given a definite integral that can be evaluated using trigonometric subtu on, we could first evaluate the corresponding indefinite integral (by changing from an integral expressed in x to one in terms of θ, then convert tox back) and then evaluate using the original limit. The following equations are very useful when evaluating integrals using trigonometric subtu on.
Trigonometric Subs tu e works on the same principles as Integra e by Subs tu e, although it can feel. If one uses Trigonometric Subs on an integrand containing √. Consider the Pythagorean identity sin2θ+cos2θ=1. A). In Exercises 27 – 32, evaluate the definite integrals by making the correct trigonometric substitution and changing the limits of the integration.
Par al Frac on Decomposi on
Example 182 Decomposition into paral fractions Perform the paral frac with decomposition of 1. Example 183 Integration using paral fractions Use paral frac on decomposition to integrate. Example 185 Integration using parallel fractions Use parallel fractions in decomposition to evaluate.
Par al Frac on Decomposi on is an important tool when dealing with ra nal functions. Fill in the blank: Par al Frac on Decomposi on is a method of rewriting functions. T/F: It is a little necessary to use polynomial division before using Par al Frac on Dekomposi on.
Hyperbolic Func ons
The following key idea summarizes many important identities regarding hyperbolic functions. Example 187 Derivatives and integrals of hyperbolic functions Evaluate the following derivatives and integrals. The following key ideas give derivatives and integrals related to inverse hyperbolic functions.
Note how inverse hyperbolic func ons can be used to solve integrals that we used to solve trigonometric substu on in Section 6.4. The hyperbolic functions are used to define points on the right side of the hyperbolax2−y2=1, as shown in Figure 6-13. How can we use the hyperbolic functions to define points on the left side of the hyperbola.
L’Hôpital’s Rule
Note that at each step where l'Hôpital's rule was applied, it was necessary: the initial limit returned the indeterminate form of "0/0." For example, if the initial limit returns 1/2, then l'Hôpital's rule does not apply. It allows the technique to be applied to the indeterminate form∞/∞ and to limit wherex approaches±∞. When faced with an indeterminate form such as 0·∞or∞−∞, we can use algebra to rewrite the limit so that l'Hôpital's rule can be applied.
Asx→ ∞, the argument of the ln term approaches ∞/∞, for which we can apply l'Hôpital's Rule. When faced with an undefined form involving a power, it helps to use the natural logarithmic function. Fill in the blanks: Quo ent rule applies tof(x) g(x) when ; l'Hôpital's rule applies when taken.
Improper Integra on
We learned Subs - tu on, which "undoes" the Chain Rule of Differences, as well as Integra on by Parts, which "undoes" the Product Rule. Example 201 Finding area: integration with respect to toys Find the area of the area enclosed by the function=√. Example 202 Find the area of a triangle Calculate the area of the areas bounded by the lines.
To approximate the surface area of a lake, shown in Figure 7.7 (a), the "length" of the lake is measured in 200-foot increments as shown in Figure 7.7 (b), where the lengths are given in hundreds of feet. Use the trapezoidal rule to approximate the area of the pictured lake, whose lengths, in hundreds of feet, are measured in 100-foot increments. Use Simpson's Rule to approximate the area of the pictured lake, whose lengths, in hundreds of feet, are measured in 200-foot increments.
Volume by Cross-Sec onal Area; Disk and Washer Methods
Find the volume of the solid formed by revolving the curvey=1/x, fromx=1 tox=2, around the dex axis. Find the volume of the solid formed by rotating the curvey=1/x, fromx=1 tox=2, around the side axis. If the outer radius of the body is R(x) and the inner radius (which defines the hole) is r(x), then it is volume.
Find the volume of the solid formed by rotating the region bounded by y = x2−2x+2 eny=2x−1 around the x-axis. Use the Disk/Washer Method to find the volume of the solid of revolution formed by the area of rotation about the axis. Use the Disk/Washer Method to find the volume of the solid of revolution formed by the area of rotation about their axis.
The Shell Method
The radius of the shell formed by the different element is the distance of xtox =3; that is, it isr(x) = 3−x. Find the volume of the solid formed by the area given in Example 210 to rotate the x-axis. Find the volume of solid formed by rotating the region bounded by y=sinx and the x-axis from x=0 tox=π around the side axis.
Use Shell's method to find the volume of a solid of revolution produced by the rotation of a region about its axis. Use Shell's method to find the volume of a solid of revolution produced by rotating a region about the x-axis. Use the shell method to find the volume of a solid of revolution produced by rotating a region about each of the given axes.
Arc Length and Surface Area
Thus the surface area of this sample frustum of the cone is approximately 2πf(xi) +f(xi+1). The surface of the solid formed by the rotation of the graph ofy= f(x), kuf(x)≥0, about the x-axis is. The surface of the solid formed by the rotation of the graph ofy= f(x) around their axis, while,b≥0, is.
Determine the surface area of the solid formed by revolvingy=sinxon[0,π] around the x-axis, as shown in Figure 7-29. Since the integral is larger on the right-hand side, we conclude that it also diverges, meaning that Gabriel's horn has an infinite area. In Exercises 13 – 20, set the integral to calculate the arc length of the function at the given interval.
Work
How much work is done to pull the rope up, where the rope has a mass of 66g/m. In order to do this, we must first decide what is measuring: it is the length of the rope that will hang or is it the amount of rope that is pulled in. For this example, we adopt the convention that this is the amount of string pulled in.
It seems to fit intui on be er; pulling up the first 10 meters of rope involves x=0 tox=10 instead of x=60 tox=50. Example 220 Doing computer work: application of variable force Consider again pulling a 60 m rope up a cliff, where the rope has a mass of 66 g/m. We want to find a height so that the work of pulling the string from a height of x=0 to a height of x = side 582.12 is half the total work.
Fluid Forces
Infinite Series
Integral and Comparison Tests
Ra o and Root Tests
![Figure 5.10: What is the area below y = x 2 on [0, 3]? The region is not a usual ge-ometric shape.](https://thumb-ap.123doks.com/thumbv2/123dok/10224568.0/26.918.61.273.110.292/figure-area-y-region-usual-ge-ometric-shape.webp)
![Figure 5.27: A graph of y = sin x on [0, π] and the rectangle guaranteed by the Mean Value Theorem.](https://thumb-ap.123doks.com/thumbv2/123dok/10224568.0/55.918.642.866.114.292/figure-graph-sin-rectangle-guaranteed-mean-value-theorem.webp)
