Calculus II: Chapter 4-6
How many antiderivatives does a given function have? What do those antiderivatives all have in common?
A function has infinitely many antiderivatives and they all have the same shape.
How do we measure the work accomplished by a varying force that moves an object a certain distance?
One uses the equation for force as the integrand, and then one then uses the increment of height as the increment of change, within a definite integral.
How do we accurately evaluate a definite integral such as ∫10e−x2dx∫01e−x2dx when we cannot use the First Fundamental Theorem of Calculus because the integrand lacks an elementary algebraic antiderivative? Are there ways to generate accurate estimates without using extremely large values of n in Riemann sums?
Yes, one can use the trapezoidal rule, and Simpson's rule by extension.
What is the meaning of the definite integral of a rate of change in contexts other than when the rate of change represents velocity?
it is the value that represents the total change which occurs over an interval given that rate of change.
How is the problem of finding distance traveled related to finding the area under a certain curve?
By finding the area under a curve, whether the curve models the change that occurs, one multiplies a quantity, by the change at each quantity, to find the total there. This means that one simply treats it like an area problem.
What are the differences among left, right, middle, and random Riemann sums?
Each one chooses a different place in the subinterval for the height. 1. Ln is when the left, i=0, most x value is chosen and then the height is given. 2. Rn is when the right most, i=1, x value is chosen and then the height is given. 3. Middle is when one chooses from the middle, found from the average of the two ends, x value in the subinterval. 4. Random is when one chooses a value from the subinterval randomly, usually used in theory.
What are improper integrals and why are they important?
Improper integrals are integrals that have some infinite component that one must approximate with a limit whether the interval or the integrand is infinite, positive or negative. These are important as they can give the overall, for everything, that occurs.
What role have integral tables historically played in the study of calculus and how can a table be used to evaluate integrals such as sqrt(a^2+u^2) du?
Integral tables have served the purpose of allowing standard integrand forms to have their antiderivatives easily found by simply substituting u values, done on the integrand, to some general form which has already been determined. Without this, trigonometric substitution would be necessary.
What role can a computer algebra system play in the process of finding antiderivatives?
A CAS can solve whatever one inputs, as it is able to perform symbolic mathematics, so it can help simplify the process and find a straightforward answer with careful human care being taken when entering answers.
How can we begin to find algebraic formulas for antiderivatives of more complicated algebraic functions?
One can begin by looking at the structure of the derivative and determining what elementary derivative rules have been applied so that they can be reversed.
How do we decide whether to integrate with respect to x or with respect to y when we try to find the area of a region?
One can decide whether to integrate with regards to x or y based on whether one can definitely have there is a constant upper and lower function.
How can we write Riemann sums in an abbreviated form?
One can simply put an index (usually the function that one wants to put in some value n number of times) write from what point it starts, and then put sigma sign (add up after inputs!)
Given the graph of a function's derivative, how can we construct a completely accurate graph of the original function?
One can simply solve the fundamental theorem of calculus equation for one point, and then plot that point for the value of b that it extends to.
How can we use definite integrals to measure the area between two curves?
One can subtract the lesser function from the greater one and this becomes the integrand, and then find intersections and these intersections are the limits of integration.
How are both of the above concepts and their corresponding use of definite integrals similar to problems we have encountered in the past involving formulas such as "distance equals rate times time" and "mass equals density times volume"?
One can take small amounts of distance or mass and then add up all these small parts which have some changing component in the integrand that requires a small interval.
How can we use a Riemann sum to estimate the area between a given curve and the horizontal axis over a particular interval?
One can use a Riemann sum to sum the form f(x)*(change in x) for a certain number of rectangles and find the total area (net or overall) of some region.
If we know the velocity of a moving body at every point in a given interval, can we determine the distance the object has traveled on the time interval?
One can, if the shape is regular, simply find the area under a curve, or, if irregular, approximate the area as the sum of triangles.
What are some typical improper integrals that we can classify as convergent or divergent?
One is the form 1/(x^p) where p>1 it converges and diverges 0 < less than or equal to 1, or P is less than one but still positive.
What adjustments do we need to make if we revolve about a line other than the x- or y-axis?
One needs to adjust the given functions in some way that reflects the new height given that this new line is some translation of the functions.
What is the statement of the Fundamental Theorem of Calculus, and how do antiderivatives of functions play a key role in applying the theorem?
The Fundamental Theorem of Calculus states that the value of the definite integral is equal to the change in position that is found by subtracting two y-values for the position function each evaluated at the endpoint of the interval. This requires that the antiderivative be found, and the endpoints plugged in, and then subtracted to find the difference which is distance traveled or change in position.
How are the errors in the Trapezoid Rule and Midpoint Rule related, and how can they be used to develop an even more accurate rule?
The Trapezoid rule has an error that is one half the negative value of the midpoint rule and this can then be used, by averaging them and eliminating some error.
How can we find the exact value of a definite integral without taking the limit of a Riemann sum?
The exact value can be found by finding the antiderivative of the index or integrand.
How do the First and Second Fundamental Theorems of Calculus enable us to formally see how differentiation and integration are almost inverse processes?
The first and second FTCs both show how the derivative, the rate of change, and the antiderivative, the total amount, are related by the process of differentiation and integration.
What is an indefinite integral and how is its notation used in discussing antiderivatives?
The indefinite integral is the general form, the antiderivative, that results from performing integration. Usually, this is the amount form. It lacks limits of integration.
How does increasing the number of subintervals affect the accuracy of the approximation generated by a Riemann sum?
The more subintervals, the more rectangles which better approximates the space as the tops approach a smooth curve like the one that needs to be approximated.
What is the total force exerted by water against a dam?
The total force exerted by water against a dam is the hydrostatic pressure multiplied by the area.
What is the Trapezoid Rule, and how is it related to left, right, and middle Riemann sums?
The trapezoidal rule is the approximation of the definite integral by using area of trapezoids. The average of the left and right Riemann sum is equivalent to this.
How does the technique of u-substitution work to help us evaluate certain indefinite integrals, and how does this process rely on identifying function-derivative pairs?
The use of u-substitution functions to reverse the chain rule, which differentiates a composite function, and returns the function back to a composite function. This needs one to select the function which has a derivative-antiderivative pair is that which is u and is then simplified.
How can a definite integral be used to measure the length of a curve?
There can be the integral of sqrt(1+(f'(x))^2)
How does the integral function A(x)=∫x1f(t)dtA(x)=∫1xf(t)dt define an antiderivative of f?
This is an integral that gives an expression with a variable end, but a constant beginning that is essentially an open ended interval. It is not evaluated for a specific interval.
What is the definition of the definite integral of a function f over the interval [a,b]?
This is the net area of a function over the integral, and it is the limit, as the number of subintervals goes to infinity, of the Riemann sum.
If velocity is negative, how does this impact the problem of finding distance traveled?
This means that one must regard this as positive, and then sum it with the other distances that are traveled.
What does the definite integral measure exactly, and what are some of the key properties of the definite integral?
This measure the net area. There are 6 key properties. 1. Interval width of zero has zero area 2. Sum rule: Add integrals of functions to get total area 3. Constant multiple: A constant can be taken out of an integral 4. Reversed bounds of integration: Just multiply the integral by a negative one to give the same sign. 5. Same function overlapping area: Just make the boundaries of integration extend over this area.
How does the method of partial fractions enable any rational function to be antidifferentiated?
This reverses the process of adding or subtracting two fractions and decomposes one into two fractions with different denominators that are linear or irreducible quadratic factors. This allows for u-substitution to be employed and then the integrand to anti differentiated.
What is the statement of the Second Fundamental Theorem of Calculus?
This states that the derivative of the integral, with respect to a variable, will give the derivative which is the integrand when in integral, indefinite, form.
Given a function f, how does the rule A(x)=∫f(t)dt define a new function A?
This states that the function is the same, when evaluated at the greatest value in an interval, as the area for some derivative as the interval is [a,x] and this defines A as an integral function account for change in position or distance traveled.
What does it mean to antidifferentiate a function and why is this process relevant to finding distance traveled?
To andifferentiate a function is to take a derivative, a equation that accounts fro how a function is changing, and then convert it back to a position function (or the function it was before the derivative, more generally). This can be used to fund the distance traveled, as once the position function is given that can. be used to solve for it.
What does it mean to say that an improper integral converges or diverges?
To say that an improper integral converges means that the value, finite, of an area, and if it diverges when the limit of the FTC antiderivative form is taken, then the area is not finite in value.
In what circumstances do we integrate with respect to y instead of integrating with respect to x?
When one rotates about the y-axis. This means that the width change in y, and then the function, if not already given as such, must be solved for y.
How can we use a definite integral to find the volume of a three-dimensional solid of revolution that results from revolving a two-dimensional region about a particular axis?
one can state that the width approaches zero, as the increment of change, and then use the other part of it, within the height as the function, as the integrand.