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AP® Calculus BC—Semester B

Time to get series-ous.

  • Course Length: 18 weeks
  • Course Type: AP
  • Category:
    • College Prep
    • Math
    • High School

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This course has been approved by the College Board, which indicates that the syllabus "has demonstrated that it meets or exceeds the curricular expectations colleges and universities have for your subject." Please contact sales@shmoop.com if you would like to add this course to your official record of AP course offerings.

It has also been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.

Craving more calculus? If you enjoyed Semester A of Advanced Calculus BC, there's no reason not to dive right into Semester B. If you're like us, it's not like you have anything better to do.

Not to mention, not taking Semester B of the course would be like leaving a bowl of mac and cheese unfinished, eating a slice of pizza while leaving the crust untouched (stuffed of course), or cutting off the movie Titanic right before they strike the iceberg.

Just in case you didn't know, Jack and Rose's relationship doesn't have much life left after that point...

Luckily, Semester B of this course ends on a substantially brighter note. 

Semester B is where the content in this course starts to pull away a bit from what's covered in AP® Calculus AB, too. Here's a rundown of topics you'll see in this course that aren't covered in our AP® Calculus AB course.

  • Advanced integration techniques, including integration by parts and partial fraction decompositions
  • Improper integrals
  • Integrals and derivatives with polar and parametric functions
  • Arc length
  • Logistic growth
  • Sequences and infinite series
  • Power series representations of differentiable functions

That's not all that's covered here, though. There's also plenty of content that's also available as part of our AP® Calculus AB offering. Here's a quick rundown of how the semester is laid out.

  • We'll start with an overview of integration, a technique for finding the area under curve. We'll learn its hobbies, interests, favorite foods, and maybe even how it takes its coffee.
  • Then we'll use everything we've learned about integration to find the lengths of curves and volumes of 3D solids. We'll even introduce polar and parametric functions, so we can produce graphs that look a lot more impressive than what we can do with plain old functions of x and y.
  • The applications of integration don't end there, though. By integrating a function's derivative over an interval, we'll see the net change in the function over that interval. That idea can be applied to a whole host of situations.
  • Then we'll do a complete 180 and see what happens when we decide to add up infinitely many numbers in an infinite series. Believe it or not, sometimes an infinite sum like this can converge to a finite number.
  • If you're wondering what that last point had to do with calculus, the final unit will bring it on home. That's where we'll represent infinitely differentiable functions as infinite series, and use this new representation to approximate function values and definite integrals.  

And remember: Advanced Calculus BC is a two-semester course. You're looking at Semester B, but if you want to check out Semester A, click here.

Technology Requirements

Seeing as this is an online course and all, a computer with internet access will be a must. A graphing calculator is required for many of the activities in this course as well. The instructions given in the activities on how to use many calculator functions apply most directly to calculators in the TI-83 and TI-84 series.

Required Skills

Did you take Semester A of the course? Do you have a solid foundation in algebra, geometry, and pre-calculus? Do you know how to use the internet? Did you remember to not put your underwear on inside out today? If you answered yes to all these questions, you're good to go with this course. Trust us, doing calculus with your underwear on inside out isn't easy.

Unit Breakdown

7 AP® Calculus BC—Semester B - Introduction to Integration

This unit introduces the next big idea in calculus: integration. We'll show you how to use integrals to find the area underneath a curve, and then how integration and differentiation are two sides of the same coin through the Fundamental Theorem of Calculus. When a theorem has a name as impressive as that, you know it's going to be important.

8 AP® Calculus BC—Semester B - Area, Volume, and Arc Length

When it comes to functions, we're used to taking in a point x, and having our function spit out another point, y. That's not the only way to get the job done, though. Polar and parametric functions provide a different way of producing graphs in the coordinate plane. We'll show you how to work with these functions, as well as how to compute derivatives and integrals with them. Then we'll leave the coordinate plane and close out the unit by generating 3D solids, and using integrals to compute their volumes.

9 AP® Calculus BC—Semester B - Further Applications of Integration

The applications of integration don't end at area, volume, and arc length. We can use integrals to find the average value of a function, as well as a function's net change over an interval. Then we'll revisit differential equations, and use integrals to solve differential equations we just couldn't crack before.

10 AP® Calculus BC—Semester B - Sequences and Series

You might remember that a sequence is just an infinite list of numbers. But what happens if we decide to add up all those numbers? The result is a series. We'll show you how to work with these guys, as well as how to tell if these infinite sums converge or diverge. It may seem counter intuitive, but it's totally possible to add up infinitely many numbers, and get a clean, finite answer.

11 AP® Calculus BC—Semester B - Power Series

This is where we link series with differentiation and integration to end the course with a bang. We'll show you how to take an infinitely differentiable function and represent it as an infinite series. You may be wondering why this is useful, but representing functions as series allows us to approximate function values and definite integrals, and make that estimate as precise as we want. We can't think of a better way to cap the course off.

Recommended prerequisites:

  • AP® Calculus BC—Semester A

    • Sample Lesson - Introduction

    Lesson 8.10: The Shell Method



    https://media1.shmoop.com/shmooc/ap-calculus/intro/shmooc_ap_calc_ab_unit9_intro_graphik_13.png
    The hair on that shell makes the milk look really unappetizing.
    (Source)

    When stranded on a desert island, what's the first thing any reasonable person would do? Well, most of us would probably do everything in our power to get off the island, but that's going to require a lot of energy and probably not end well. The best thing to do is hunker down and wait for someone to come to our aid. Hunkering down is going to involve finding food, shelter, and a source of water.

    Water is probably the most important thing we'll need, so we're going to want to start there first. Unfortunately, desert islands are kind of known for not having fresh water, so we may need to crack open a coconut or two to get at the milk. By the way, that's not as easy as it sounds.

    And since we're stranded on this island, we don't want to waste any resources, including those leftover coconut shells. Sure, we could use them to fashion a few stylish coconut shell bras, but with all this free time to do calculus while we await our rescue, they might also make nice cross-sections for solids of revolution, right alongside disks and washers. The only way we'll know is if we try.

    Sample Lesson - Reading

    Reading 8.8.10: A Shell, a Cylinder, Same Thing

    If the disk method can find the volume of a solid using disks, and the washer method can find volume using washers, then it's probably safe to assume that the shell method finds volume using shells. Actually it might be more appropriate to call it the cylinder method, since the cross-sections are going to end up looking a lot more like cylinders than shells, but the shell method just has a certain ring to it. Like, no one really knows why Batman's sidekick is called Robin, but we can all agree that "Batman and Robin" sounds a lot better than "Batman and Batboy".

    To get the low down on how to use the shell method, check out this reading. The shell method is usually the hardest method for finding volumes of solids, because it can get difficult to imagine what the cross-sections look like before setting up our integral.

    The other complication is finding the radius of each shell. In the example in the reading, the axis of revolution was the y-axis, and the edge of each shell was "x" so that was the radius as well. But if we revolve a region around a different line, that's going to skew the radius. For example, take a look at the region below:

    Let's imagine we revolve this region around the line x = 1, and try to find the volume of the resulting solid using the shell method. The black bar represents the location and height of some random shell. The height of each shell is going to be ln(x – 1), but the radius of each shell extends from the edge of each shell to the line x = 1, instead of going all the way to the y-axis. As such, the radius will be x – 1. And since the region starts at x = 2 and ends at x = 4, those will be the limits of integration.

    So putting this all together, the volume of the region would be

    And that's the shell method, in a nutshell. Sorry, we couldn't resist.

    Recap

    The shell method gives us a way of finding the volume of solids of revolution where the cross-sections look like cylinders. We can find the volume using the integral where r is the radius of the cylinder, and h is the height.

    Sample Lesson - Activity

    1. A region bounded by y = sin x + 5, y = 0, x = 4, and x = 7 is rotated about the y-axis. Which integral gives the volume of the solid of revolution?

    2. A region is bounded by the y-axis, x = 2y2 + 8, y = 2, and y = 6. For which axis of rotation would we use the shell method to find the volume of the solid of revolution, if we integrate with respect to y?

    3. Given a region bounded by the x-axis, y = x2 – 3, x = 1, and x = 2, for which axis of rotation would we use shells to find the volume of the solid of revolution, if we integrate with respect to x?

    4. What is the volume of the solid of revolution formed when the region bounded by y = x2 + 2x, the coordinate axes, and x = 3 is rotated about the y-axis?

    5. What is the volume of the solid of revolution when the region bounded by f(x) = x2 + 3 and g(x) = x + 3 is rotated about the y-axis?

    6. What is the volume of the solid of revolution formed when the region bounded by f(x) = -x2 + 13x – 36 and g(x) = -6 is rotated about the line x = 14?

    7. What is the volume of the solid of revolution formed when the region bounded by x = 4, x = 16, y = -1, and  is rotated about the line x = -2?

    8. What is the volume of the solid of revolution when the region bounded by x = y and x = y2 is rotated about the line y = -3?

    9. What is the volume of the solid formed when the region bounded by x = y2 + 2y + 1, the x-axis, and x = 16 in the first quadrant is rotated about the x-axis?

    10. What is the relationship between the axis of revolution and which variable to integrate with respect to when using the shell method?

    11. What is a solid of revolution with shell-like cross sections where the height of each shell and the radius are the same?

    12. Take a fixed region R, that's been rotated about an axis to form a solid of revolution. The larger the radius of each shell-like cross section of the solid, the larger the volume of the solid of revolution will be.

    13. When rotating a region about the x-axis, the radius of the shell is x.

    14. Both the radius and height of a shell must be in terms of the same variable in order to use the shell method.

    15. When using the shell method, if the height of each cross section of the solid is the same, then the volume is .