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Unit 8: Applications of integration
About this unit.
Definite integrals are all about the accumulation of quantities. Let's see how they are applied in order to solve various kinds of problems.
Finding the average value of a function on an interval
- Average value over a closed interval (Opens a modal)
- Calculating average value of function over interval (Opens a modal)
- Mean value theorem for integrals (Opens a modal)
- Average value of a function Get 3 of 4 questions to level up!
Connecting position, velocity, and acceleration functions using integrals
- Motion problems with integrals: displacement vs. distance (Opens a modal)
- Analyzing motion problems: position (Opens a modal)
- Analyzing motion problems: total distance traveled (Opens a modal)
- Motion problems (with definite integrals) (Opens a modal)
- Worked example: motion problems (with definite integrals) (Opens a modal)
- Average acceleration over interval (Opens a modal)
- Analyzing motion problems (integral calculus) Get 3 of 4 questions to level up!
- Motion problems (with integrals) Get 3 of 4 questions to level up!
Using accumulation functions and definite integrals in applied contexts
- Area under rate function gives the net change (Opens a modal)
- Interpreting definite integral as net change (Opens a modal)
- Worked examples: interpreting definite integrals in context (Opens a modal)
- Analyzing problems involving definite integrals (Opens a modal)
- Worked example: problem involving definite integral (algebraic) (Opens a modal)
- Interpreting definite integrals in context Get 3 of 4 questions to level up!
- Analyzing problems involving definite integrals Get 3 of 4 questions to level up!
- Problems involving definite integrals (algebraic) Get 3 of 4 questions to level up!
Finding the area between curves expressed as functions of x
- Area between a curve and the x-axis (Opens a modal)
- Area between a curve and the x-axis: negative area (Opens a modal)
- Area between curves (Opens a modal)
- Worked example: area between curves (Opens a modal)
- Composite area between curves (Opens a modal)
- Area between a curve and the x-axis Get 3 of 4 questions to level up!
- Area between two curves given end points Get 3 of 4 questions to level up!
- Area between two curves Get 3 of 4 questions to level up!
Finding the area between curves expressed as functions of y
- Area between a curve and the 𝘺-axis (Opens a modal)
- Horizontal area between curves (Opens a modal)
- Horizontal areas between curves Get 3 of 4 questions to level up!
Finding the area between curves that intersect at more than two points
- No videos or articles available in this lesson
- Area between curves that intersect at more than two points (calculator-active) Get 3 of 4 questions to level up!
Volumes with cross sections: squares and rectangles
- Volume with cross sections: intro (Opens a modal)
- Volume with cross sections: squares and rectangles (no graph) (Opens a modal)
- Volume with cross sections perpendicular to y-axis (Opens a modal)
- Volumes with cross sections: squares and rectangles (intro) Get 3 of 4 questions to level up!
- Volumes with cross sections: squares and rectangles Get 3 of 4 questions to level up!
Volumes with cross sections: triangles and semicircles
- Volume with cross sections: semicircle (Opens a modal)
- Volume with cross sections: triangle (Opens a modal)
- Volumes with cross sections: triangles and semicircles Get 3 of 4 questions to level up!
Volume with disc method: revolving around x- or y-axis
- Disc method around x-axis (Opens a modal)
- Generalizing disc method around x-axis (Opens a modal)
- Disc method around y-axis (Opens a modal)
- Disc method: revolving around x- or y-axis Get 3 of 4 questions to level up!
Volume with disc method: revolving around other axes
- Disc method rotation around horizontal line (Opens a modal)
- Disc method rotating around vertical line (Opens a modal)
- Calculating integral disc around vertical line (Opens a modal)
- Disc method: revolving around other axes Get 3 of 4 questions to level up!
Volume with washer method: revolving around x- or y-axis
- Solid of revolution between two functions (leading up to the washer method) (Opens a modal)
- Generalizing the washer method (Opens a modal)
- Washer method: revolving around x- or y-axis Get 3 of 4 questions to level up!
Volume with washer method: revolving around other axes
- Washer method rotating around horizontal line (not x-axis), part 1 (Opens a modal)
- Washer method rotating around horizontal line (not x-axis), part 2 (Opens a modal)
- Washer method rotating around vertical line (not y-axis), part 1 (Opens a modal)
- Washer method rotating around vertical line (not y-axis), part 2 (Opens a modal)
- Washer method: revolving around other axes Get 3 of 4 questions to level up!
The arc length of a smooth, planar curve and distance traveled
- Arc length intro (Opens a modal)
- Worked example: arc length (Opens a modal)
- Arc length Get 3 of 4 questions to level up!
Calculator-active practice
- Contextual and analytical applications of integration (calculator-active) Get 3 of 4 questions to level up!
COMMENTS
To compute the indefinite integral R R(x)dx, we need to be able to compute integrals of the form Z a (x n ) dx and Z bx+c (x2 + x+ )m dx: Those of the first type above are simple; a substitution u= x will serve to finish the job. Those of the second type can, via completing the square, be reduced to integrals of the form bx+c (x 2+a)m dx.
AP Calculus AB - Worksheet 46 Integration of Inverse Trigonometric Functions Evaluate each integral. 1. dx ³ x2 9 2. 9r2 1 r3 ³ dr 3. ³cos 3 4z dz 4. dx ³ sin2 3x 5. 2 x 6cos 2 sin t dt t ³ 7. 6. x 2 1 ³ dx 40 ³ x 25 8. dx 1 4x2 ³ 9. 2 2 41 y dy ³ y 10. dx 9 x2 ³ Answers: 1. 1 3 § ©¨ · ¹¸ tan 1 x ©¨ 3 · ¹¸ C 2. 6 1 r3 C 3 ...
AP Calculus BC - Worksheet 50 Definite Integrals as Riemann Sums In the chart below, either a definite integral or a limit of a Riemann sum has been provided. Fill in the box with the corresponding missing information Definite Integral Limit of Riemann Sum 1 6 0 ³ 21x dx 2 lim 2 3 2 1 n 55 n i i o nn f ªº§· «»¨¸ «»¬¼©¹ ¦ 3 ...
AP Calculus BC - Worksheet 41 Integration by u-Substitution Evaluate the indefinite integral by using the given substitution. 1) ³cos 6 ; 6x dx u x 8 2) ³63 9 7 ; 9 7x dx u x 3) ³28 7 ; 7r r dr u r6 7 7 Use substitution to find the indefinite integral. 4) ³12 4 8 2 y y y y dy4 2 3 2 sin 8 9 2 5) 5 53 dx x ³ 6) ³ z dz 7) 14 ln x dx ³ x 8)
Q H LA 3l 9l V QrXiBgkh zt3sV er 2eos Qesr1v pesd g.B Y ZMNaLd YeM Kw ni yt nhE oI9n Qffi zn hiwtLeK lC Kaml2c9uvlduAsV.3 Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Integration by Parts Date_____ Period____ Evaluate each indefinite integral using integration by parts.
AP Calculus-AB worksheets by topics Fu n c t i o n s , L i mi t s , & Co n t i n u i t y D i f f e re n t i a t i o n 1. I n te re s t i n g G ra p h s - A few equations to graph that have interesting (and hidden) features. pdf 2. Fu n c t i o n s - Properties of functions and the Rule of Four (equations, tables, graphs, and words).
Course: AP®︎/College Calculus BC > Unit 6. Lesson 13: Using integration by parts. Integration by parts intro. Integration by parts: ∫x⋅cos (x)dx. Integration by parts: ∫ln (x)dx. Integration by parts: ∫x²⋅𝑒ˣdx. Integration by parts: ∫𝑒ˣ⋅cos (x)dx. Integration by parts. Integration by parts: definite integrals.
AP CALCULUS Worksheet - Evaluating Definite Integrals. 2 12) What is the exact area of the region between y x and the x-axis , over the interval [0, 1]? ... properties Of definite integrals to evaluate each expression. 112 f (x) dr = —4, 115 f (x) clx = 6,
Some integrals like sin(x)cos(x)dx have an easy u-substitution (u = sin(x) or cos(x)) as the 'u' and the derivative are explicitly given. Some like 1/sqrt(x - 9) require a trigonometric ratio to be 'u'. Some other questions make you come up with a completely (seemingly) irrelevant 'u' which actually simplifies the integral.
The definite integral is an important tool in calculus. It calculates the area under a curve, or the accumulation of a quantity over time. Riemann sums allow us to approximate integrals, while the fundamental theorem of calculus reveals how they connect to derivatives. ... AP Calculus AB solved free response questions from past exams. Unit 10 ...
AP Calculus AB Integration Practice Worksheet 1) 𝐚 𝒙+𝒄 2) ( + 𝒙+𝒙 ) +𝒄 3) √( +𝒙 )+𝒄 4) 𝐢 (𝒙 )+𝒄 5)
Curriculum Module: Calculus: Fu nctions Defined by Integrals 2. Prerequisites: Students will need to: • Understand the definite integral as signed area (before Worksheet 1). • Understand the analysis of functions using first and second derivatives (before Worksheet 1).
AP Calculus BC Worksheet 2.5 Evaluate the definite integral. No calculator. 1. 2. 3. 4. 5. 6. 7. Use a calculator (MATH #9) to evaluate the definite integral.
2, f'(2) = 5, f(2) = 10 — sin t) dt (92 + sec2 e) dB f'(x) = 4x,f(0) = 6 h'(t) = 8t3 + 5, /1(1) = — (2 sin x + 3 cosx)dx (1 — csc t cot t) dt
Mixed Integration Worksheet Part I: For each integral decide which of the following is needed: 1) substitution, 2) algebra or a trig identity, 3) nothing needed, or 4) can't be done by the techniques in Calculus I. Then evaluate each integral (except for the 4th type of course). A.∫(xdx3 +1) 23( ) 4
6. Consider the function f that is continuous in the interval [-5, 5] and for which 4 5 0 ³f x dx. Evaluate each integral. a) ³>f b) x @dx 5 0 3 ³f x dx 3 2 2 (Hint: assume the graph for f(x) is known, and sketch the graph of f(x+2)) c) ³f x dx 5 5 (f is even.)d) ³f x dx 5 5 (f is odd.)In 7-10, determine whether the statement is true or false.If it is false, explain why or give an
AP Calculus BC solved exams. Unit 12. AP®︎ Calculus BC Standards mappings. Course challenge. Test your knowledge of the skills in this course. Start Course challenge. ... Analyzing motion problems (integral calculus) Get 3 of 4 questions to level up! Motion problems (with integrals) ...
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4 The integrand is a rational function, so we use the method of partial fractions. The denominator factors to. ( x − 2 ) , so the partial fraction decomposition takes the form. + = B 4. − ( − . This lets us determine. x 2 x x 2 ) ( A + B ) x − 2 A = 4, meaning A + B = 0 and − 2 A = 4.
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6. Consider the function f that is continuous in the interval [-5, 5] and for which 4 5 0 f x dx³ Evaluate each integral. a) ³ f> x dx 5 0 3 b) ³f x dx 3 2 2 (Hint: assume the graph for f(x) is known, and sketch the graph of f(x+2)) c) ³f x dx 5 5 (f is even.)d) (f is odd.)In 7-10, determine whether the statement is true or false.If it is false, explain why or give an
Differential Equations. Slope Fields. Introduction to Differential Equations. Separable Equations. Exponential Growth and Decay. Free Calculus worksheets created with Infinite Calculus. Printable in convenient PDF format.
Download free-response questions from past exams along with scoring guidelines, sample responses from exam takers, and scoring distributions. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at [email protected]. The ...