![]() Desired differential equations can be entered & many more features can be explored (e.g. See one tangent segment “pilot” the field & create the graph of f. This video demo shows the disk/washer technique, but the same file also has pages presenting an identical development of the shell method.įirst, develop the meaning of a slope field by gliding a dynamic “slope column” across the graph of f ‘. Then sweep a cross-section through the solid which would give the exact volume via integration. Use any function, any horizontal or vertical axis of revolution, any partitioning, any interval. Third, revolve the entire region (infinitely many, infinitely thin rectangles). Second, revolve a discrete number of rectangles determined by the curve which would approximate the solid’s volume. Volumes by Revolution (Calculus In Motion)īuild this concept in 3 stages. Afterwards, students are able to transfer what they have learned to new bases and new cross-sections. This classic problem is one of several that walk through the visualization of these difficult shapes one step at a time. Change the domain, morph the function, or use the examples on the other pages of the file. Morph the original curve to see the effects on the other components.Ĭhoose 1 to 80 subdivisions of interval and approximate the integral using rectangles for left sums, right sums, or midpoints or use trapezoids for the Trapezoidal Rule. Pages cover polynomials, trig, exp, ln, parametric, polar, and any curve of your own choosing. ![]() Interact with graphs including sliding tangents, 1st and 2nd derivatives, and the integral as an accumulation of heights. Graph f, Tangent, Derivative, Integral (Calculus In Motion) ![]() More Videos of Calculus In Motion in use (single animation files) Pre-Algebra, Algebra 1, Geometry, Algebra 2, Trigonometry, Pre-Calculus * * * (Subsequent AP Exam FRQ animations through 2022 have been written in the same manner as shown in the 6 years of videos posted above.)ĪLGEBRA IN MOTION ™ (GEOMETRY included too!)Ģ24 animations of key topics in: ( all in one set) Integrals (def, Riemann Sums, area, MVT, arc length)ĪB/BC Calculus Exam Free Response Questions - 2019, 2018, 2017, 2016, 2015, 2014 AB/BC Calculus Exam Free Response Question animations continue to be added each year and continue to be constructed in the same way as the 6 years of video samples below.Ģ81 animations of nearly every concept in a first year calculus course including animations of 2022-1997 released AP Exam AB & BC Free Response QuestionsĬLICK BELOW for a quick VIDEO peek at a cluster of animations for …ĭerivatives (def, graphs, rel rates, MVT, slope fields, etc) In the actual animations, a Motion Controller can be accessed to control the speed of any animation as desired. The actual animations run smoother and sharper on their own. Screen capture software to create videos noticeably affects the running of the animations. These videos provide just a glimpse of the actual power and capabilities of the animations. So, to reveal as much information as possible, features are displayed very rapidly – MUCH faster than “classroom pace”. Each video has been restricted to be under 3 minutes (most just 1 or 2 minutes). The videos on this page show quick views of a few of the animations in use. They are the actual animation files YOU get to interact with yourself. ![]() So to summarize the purpose of the disk method, it is because the alternative, multidimensional integration, gets pretty nasty.PLEASE NOTE: Calculus In Motion and Algebra In Motion animations are NOT videos. It is because of this symmetry that methods of multivariable calculus allow the multidimensional integral to be simplified to a single-variable integral. The disk method (along with the shell method) are shortcuts used to compute volumes of objects, that have a certain rotational invariance to them, such as the sphere, torus, bowl, or baseball bat. You will, however, need to get the hang of thinking in three dimensions. However, if you wish to look ahead, you can find that just as the integral (as you know it) can be used to compute the area bounded by the curve y = f( x), a two-dimensional integral can be used to compute the volume of a solid bounded by the surface z = F( x, y). Can you write down that sum?īecause multidimensional integration, which is is what is generally necessary to compute volumes of solids, is not an appropriate topic for an introductory integral calculus course. By the way, this image was used on an old exam for students to write down a four-term Riemann sum that approximates the volume of the bat. The disk method can be used to compute the volume of this baseball bat. If you rotated this region around the x-axis, you would obtain a baseball bat. Short answer: To compute the volume of some solids.Ĭonsider the region graphed here, bounded the function graphed in brown.
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