Version #1 The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020. BC Topics are listed, but there will be no lessons available for SY 2020-2021Unit 0 - Calc Prerequisites (Summer Work) 0.1 Summer Packet Unit 1 - Limits and Continuity 1.1 Can Change Occur at an Instant? 1.2 Defining Limits and Using Limit Notation 1.3 Estimating Limit Values from Graphs 1.4 Estimating Limit Values from Tables 1.5 Determining Limits Using Algebraic Properties (1.5 includes piecewise functions involving limits) 1.6 Determining Limits Using Algebraic Manipulation 1.7 Selecting Procedures for Determining Limits (1.7 includes rationalization, complex fractions, and absolute value) 1.8 Determining Limits Using the Squeeze Theorem 1.9 Connecting Multiple Representations of Limits Mid-Unit Review - Unit 1 1.10 Exploring Types of Discontinuities 1.11 Defining Continuity at a Point 1.12 Confirming Continuity Over an Interval 1.13 Removing Discontinuities 1.14 Infinite Limits and Vertical Asymptotes 1.15 Limits at Infinity and Horizontal Asymptotes 1.16 Intermediate Value Theorem (IVT) Review - Unit 1 Unit 2 - Differentiation: Definition and Fundamental Properties 2.1 Defining Average and Instantaneous Rate of Change at a Point 2.2 Defining the Derivative of a Function and Using Derivative Notation (2.2 includes equation of the tangent line) 2.3 Estimating Derivatives of a Function at a Point 2.4 Connecting Differentiability and Continuity 2.5 Applying the Power Rule 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple (2.6 includes horizontal tangent lines, equation of the normal line, and differentiability of piecewise) 2.7 Derivatives of cos(x), sin(x), e^x, and ln(x) 2.8 The Product Rule 2.9 The Quotient Rule 2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x) Review - Unit 2 Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions 3.1 The Chain Rule 3.2 Implicit Differentiation 3.3 Differentiating Inverse Functions 3.4 Differentiating Inverse Trigonometric Functions 3.5 Selecting Procedures for Calculating Derivatives 3.6 Calculating Higher-Order Derivatives Review - Unit 3
Unit 4 - Contextual Applications of Differentiation 4.1 Interpreting the Meaning of the Derivative in Context 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration 4.3 Rates of Change in Applied Contexts Other Than Motion 4.4 Introduction to Related Rates 4.5 Solving Related Rates Problems 4.6 Approximating Values of a Function Using Local Linearity and Linearization 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms Review - Unit 4 Unit 5 - Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 5.3 Determining Intervals on Which a Function is Increasing or Decreasing 5.4 Using the First Derivative Test to Determine Relative Local Extrema 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema 5.6 Determining Concavity of Functions over Their Domains 5.7 Using the Second Derivative Test to Determine Extrema Mid-Unit Review - Unit 5 5.8 Sketching Graphs of Functions and Their Derivatives 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative (5.9 includes a revisit of particle motion and determining if a particle is speeding up/down.) 5.10 Introduction to Optimization Problems 5.11 Solving Optimization Problems 5.12 Exploring Behaviors of Implicit Relations Review - Unit 5 Unit 6 - Integration and Accumulation of Change 6.1 Exploring Accumulation of Change 6.2 Approximating Areas with Riemann Sums 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation 6.4 The Fundamental Theorem of Calculus and Accumulation Functions 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Mid-Unit Review - Unit 6 6.6 Applying Properties of Definite Integrals 6.7 The Fundamental Theorem of Calculus and Definite Integrals 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 6.9 Integrating Using Substitution 6.10 Integrating Functions Using Long Division and Completing the Square 6.11 Integrating Using Integration by Parts (BC topic) 6.12 Integrating Using Linear Partial Fractions (BC topic) 6.13 Evaluating Improper Integrals (BC topic) 6.14 Selecting Techniques for Antidifferentiation Review - Unit 6 Unit 7 - Differential Equations 7.1 Modeling Situations with Differential Equations 7.2 Verifying Solutions for Differential Equations 7.3 Sketching Slope Fields 7.4 Reasoning Using Slope Fields 7.5 Euler's Method (BC topic) 7.6 General Solutions Using Separation of Variables 7.7 Particular Solutions using Initial Conditions and Separation of Variables 7.8 Exponential Models with Differential Equations 7.9 Logistic Models with Differential Equations (BC topic) Review - Unit 7 Unit 8 - Applications of Integration 8.1 Average Value of a Function on an Interval 8.2 Position, Velocity, and Acceleration Using Integrals 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts 8.4 Area Between Curves (with respect to x) 8.5 Area Between Curves (with respect to y) 8.6 Area Between Curves - More than Two Intersections Mid-Unit Review - Unit 8 8.7 Cross Sections: Squares and Rectangles 8.8 Cross Sections: Triangles and Semicircles 8.9 Disc Method: Revolving Around the x- or y- Axis 8.10 Disc Method: Revolving Around Other Axes 8.11 Washer Method: Revolving Around the x- or y- Axis 8.12 Washer Method: Revolving Around Other Axes 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC topic) Review - Unit 8 Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics) 9.1 Defining and Differentiating Parametric Equations 9.2 Second Derivatives of Parametric Equations 9.3 Arc Lengths of Curves (Parametric Equations) 9.4 Defining and Differentiating Vector-Valued Functions 9.5 Integrating Vector-Valued Functions 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions 9.7 Defining Polar Coordinates and Differentiating in Polar Form 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve 9.9 Finding the Area of the Region Bounded by Two Polar Curves Review - Unit 9 Unit 10 - Infinite Sequences and Series (BC topics) 10.1 Defining Convergent and Divergent Infinite Series 10.2 Working with Geometric Series 10.3 The nth Term Test for Divergence 10.4 Integral Test for Convergence 10.5 Harmonic Series and p-Series 10.6 Comparison Tests for Convergence 10.7 Alternating Series Test for Convergence 10.8 Ratio Test for Convergence 10.9 Determining Absolute or Conditional Convergence 10.10 Alternating Series Error Bound 10.11 Finding Taylor Polynomial Approximations of Functions 10.12 Lagrange Error Bound 10.13 Radius and Interval of Convergence of Power Series 10.14 Finding Taylor Maclaurin Series for a Function 10.15 Representing Functions as a Power Series Review - Unit 8 | Version #2 The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day. Lessons and packets are longer because they cover more material.Unit 0 - Calc Prerequisites (Summer Work) 0.1 Things to Know for Calc 0.2 Summer Packet 0.3 Calculator Skillz Unit 1 - Limits 1.1 Limits Graphically 1.2 Limits Analytically 1.3 Asymptotes 1.4 Continuity Review - Unit 1 Unit 2 - The Derivative 2.1 Average Rate of Change 2.2 Definition of the Derivative 2.3 Differentiability [Calculator Required] Review - Unit 2 Unit 3 - Basic Differentiation 3.1 Power Rule 3.2 Product and Quotient Rules 3.3 Velocity and other Rates of Change 3.4 Chain Rule 3.5 Trig Derivatives Review - Unit 3 Unit 4 - More Deriviatvies 4.1 Derivatives of Exp. and Logs 4.2 Inverse Trig Derivatives 4.3 L'Hopital's Rule Review - Unit 4 Unit 5 - Curve Sketching 5.1 Extrema on an Interval 5.2 First Derivative Test 5.3 Second Derivative Test Review - Unit 5 Unit 6 - Implicit Differentiation 6.1 Implicit Differentiation 6.2 Related Rates 6.3 Optimization Review - Unit 6 Unit 7 - Approximation Methods 7.1 Rectangular Approximation Method 7.2 Trapezoidal Approximation Method Review - Unit 7 Unit 8 - Integration 8.1 Definite Integral 8.2 Fundamental Theorem of Calculus (part 1) 8.3 Antiderivatives (and specific solutions) Review - Unit 8 Unit 9 - The 2nd Fundamental Theorem of Calculus 9.1 The 2nd FTC 9.2 Trig Integrals 9.3 Average Value (of a function) 9.4 Net Change Review - Unit 9 Unit 10 - More Integrals 10.1 Slope Fields 10.2 u-Substitution (indefinite integrals) 10.3 u-Substitution (definite integrals) 10.4 Separation of Variables Review - Unit 10 Unit 11 - Area and Volume 11.1 Area Between Two Curves 11.2 Volume - Disc Method 11.3 Volume - Washer Method 11.4 Perpendicular Cross Sections Review - Unit 11 |
The AP Calculus Problem Book Publication history: First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005. Calculus Volume of Solids with Known Cross Sections Name ©` k2R0i1J8E cK`uatjae NSdodfLtew`arQew `LL^Cf.r l KANlaln frMihgahBtTsq ErZewsoeerLvTeUdk.-1-For each problem, find the volume of the specified solid. 1) The base of a solid is the region enclosed by y = - x2 4 + 1 and y = 0. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Home Calendar Zoom Q1 Q2 Q3 Q4 Review FRQ from AP Smacmath 7.8 Volumes with Cross Sections. 7.8 Volumes with Cross Sections. Math AP®︎/College Calculus AB Applications of integration Volumes with cross sections: squares and rectangles Volumes with cross sections: squares and rectangles AP.CALC: CHA‑5 (EU), CHA‑5.B (LO), CHA‑5.B.1 (EK).
We have seen how to find the volume that is swept out by an area between two curves when the area is revolved around an axis.On this page we will explore volumes where the cross section is known, but isn't generated by revolution.
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See About the calculus applets for operating instructions. |
1. Square on side
The applet initially shows the yellow region bounded by f (x) = x +1 and g(x) = x² from 0 to 1. This is the base of a solid which has square cross sections when sliced perpendicular to the x-axis (i.e., one side of each square lies in the yellow region). Move the x slider to move a representative slice about the region, noticing that the size of the square changes. The two ends are also shown in light gray.
Note that the slice is sticking up from the screen, and the perspective causes it to look like a rectangle. The volume of one of these square slices with thickness dx and side length s is just the area of the square times dx, or s²dx. But s is just the distance between the two curves for a given x, or s = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41/30 or about 1.367.
2. Square on diagonal
Select the second example from the drop down menu, showing the same region. This time the cross sections (when sliced perpendicular to the x-axis) are also squares, but the diagonal of the square lies on the region. This means that the square sticks up out of the screen and also down below the screen. Move the x slider to move a representative slice about the region, noticing that the size of the square changes. The volume of one of these square slices with thickness dx and diagonal length d is just the area of the square times dx, or d²/2dx. But d is just the distance between the two curves for a given x, or d = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41/60 or about 0.683.
3. Semicircle
Select the third example from the drop down menu. This time the cross sections (when sliced perpendicular to the x-axis) are semicircles with the diameter lying on the yellow region. This means that the slice sticks up out of the screen. Move the x slider to move a representative slice about the region, noticing that the size of the slice changes.The volume of one of these slices with thickness dx and diameter length d is just the area of the semicircle times dx, or π(d/2)²/2dx. But d is just the distance between the two curves for a given x, or d = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41π/240 or about 0.537.
7.8 Volumes With Cross Sectionsap Calculus 14th Edition
4. Circle
Select the fourth example from the drop down menu. This time the cross sections (when sliced perpendicular to the x-axis) are circles with the diameter lying on the yellow region. This means that the slice sticks up out of the screen and down below the screen. Move the x slider to move a representative slice about the region, noticing that the size of the slice changes. The volume of one of these slices with thickness dx and diameter length d is just the area of the circle times dx, or π(d/2)²dx. But d is just the distance between the two curves for a given x, or d = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41π/120 or about 1.073. Note that this is a different solid than one generated by revolution about an axis; in this case there is no straight-line axis.
5. Equilateral triangle
Select the fifth example from the drop down menu. This time the cross sections (when sliced perpendicular to the x-axis) are equilateral triangles with one side lying on the yellow region. This means that the slice sticks up out of the screen. Move the x slider to move a representative slice about the region, noticing that the size of the slice changes. The volume of one of these slices with thickness dx and side length s is just the area of the triangle times dx, or But s is just the distance between the two curves for a given x, or s = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41√3/120 or about 0.592.
6. Right isosceles triangle on hypotenuse
Select the sixth example from the drop down menu. This time the cross sections (when sliced perpendicular to the x-axis) are right isosceles triangles with the hypotenuse lying on the yellow region. This means that the right angle corner sticks up out of the screen. Move the x slider to move a representative slice about the region, noticing that the size of the slice changes. The volume of one of these slices with thickness dx and hypotenuse length h is just the area of the triangle times dx, or h²/4dx. But h is just the distance between the two curves for a given x, or h = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41/120 or about 0.342.
7. Right isosceles triangle on leg
Select the seventh example from the drop down menu. This time the cross sections (when sliced perpendicular to the x-axis) are right isosceles triangles with one leg lying on the yellow region. This means that the other leg and hypotenuse stick up out of the screen. Move the x slider to move a representative slice about the region, noticing that the size of the slice changes. The volume of one of these slices with thickness dx and leg length l is just the area of the triangle times dx, or l²/2dx. But l is just the distance between the two curves for a given x, or l = x +1 - x². So the integral which sums up all these slices is just We will leave it as an exercise for the reader to show that this is 41/60 or about 0.683.
7.8 Volumes With Cross Sectionsap Calculus Solver
8. Square. Slices horizontal
7.8 Volumes With Cross Sectionsap Calculus Calculator
Select the eighth example. Here the functions are functions of y instead of x and the slices are taken perpendicular to the y axis. Initially the cross section is a square. Move the y slider to move a representative slice about the region, noticing that the size of the square changes. The integral which sums up all these slices is just As you would expect (since the region is the same as example 1, just with x and y flipped), the area is the same as in example 1. You can use the choice box to select other cross section shapes.
Explore
Select the ninth example. This lets you enter your own functions, limits of integration, cross section type, and whether x and y are swapped (remember to define your functions with y if you click the inverse box).
Other 'Applications of Integration' topics
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