Pop quiz! A ball of radius 11 has a round hole of radius 5 drilled through its center. Find the volume of the resulting solid, using integrals (Scotto).

If you are having trouble solving that problem, now is an excellent time to watch Educator.com’s instructional videos on AP Calculus. You’ll learn the basics from the ground up, starting with how calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Later lessons go in-depth to make sure you learn exactly what you need to know in order to confidently ace the AP Calculus exam.

How do I prepare for the course?

Most students have already taken three or four years of secondary mathematics designed for college-bound students. Courses in algebra, geometry, trigonometry, analytic geometry, and elementary functions will give you the fundamentals needed to do well in calculus. Speak with an Academic Counselor or AP Coordinator at your school to make sure you’ve taken the right prerequisites.

According to the College Board, by the end of the AP Calculus course you should be able to:

• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal.

• Understand the meaning of the derivative in terms of a rate of change and local linear approximation and use derivatives to solve a variety of problems.

• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and use integrals.

• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

• Communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

• Model a written description of a physical situation with a function, a differential equation, or an integral.

• Use technology to help solve problems, experiment, interpret results, and verify conclusions.

• Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

• Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

Now back to that burning ball of radius. You can find the answer using cylindrical coordinates or spherical coordinates. For the sake of time and your attention span, let’s use cylindrical coordinates. The area of a circle would be pi(r^2 – 5^2). The endpoints of the integral can be found by computing sqrt(11^2 – 5^2). This is the same as sqrt(121-25) = sqrt(96), and the limits would be -sqrt(96) to sqrt(96).

Is that what you got?

As for AP Calculus BC, it’s a little more advanced than AP Calculus AB. Like Calculus AB, Calculus BC is also concerned with developing your understanding of the concepts of calculus and providing experience with its methods and applications. The course includes all topics covered in Calculus AB plus additional topics. Like the AB class, you should have already taken four years of secondary mathematics designed for college-bound students. Courses in algebra, geometry, trigonometry, analytic geometry, and elementary functions will give you the fundamentals needed to do well in calculus. Upon successful completion of the exam, you will qualify for placement and credit in a course that is one course beyond that granted for Calculus AB.

According to the College Board, by the end of the course you should be able to:

• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal.

• Understand the meaning of the derivative in terms of a rate of change and local linear approximation and use derivatives to solve a variety of problems.

• Understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change and use integrals.

• Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

• Communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

• Model a written description of a physical situation with a function, a differential equation, or an integral.

• Use technology to help solve problems, experiment, interpret results, and verify conclusions.

• Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.

• Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

What about the exams?

After you have mastered the topics mentioned above, you will take a 3 hours and 15 minutes test on the concepts and functions of calculus. Each exam consists of two sections. The first section is a multiple-choice section testing proficiency in a wide variety of topics. It consists of 45 questions in 105 minutes. Part A of the multiple-choice section (28 questions in 55 minutes) does not allow the use of a calculator. Part B of the multiple-choice section (17 questions in 50 minutes) contains some questions for which a graphing calculator is required such as the TI-98. The second section is a free-response section that requires you to demonstrate the ability to solve problems involving a more extended chain of reasoning.

This information is meant to get you started while you research AP courses. There is a lot more to learn about the classes and tests. For more information on the AP Calculus BC and AB exams visit https://www.collegeboard.com/student/testing/ap/sub_calbc.html