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Unit 8: Applications of integration | Single Variable Calculus | Project Ulim
Math
Single Variable Calculus
Unit 8: Applications of integration
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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
Learn
Average value over a closed interval
Average value: intuition behind dividing by the length of the interval
Calculating average value of function over interval
Mean value theorem for integrals
Finding the average height of a semicircle with radius 1
Practice
Average value of a function
Not started
Connecting position, velocity, and acceleration functions using integrals
Learn
Motion problems with integrals: displacement vs. distance
Analyzing motion problems: position
Analyzing motion problems: total distance traveled
Motion problems (with definite integrals)
Using accumulation functions and definite integrals in applied contexts
Learn
Area under rate function gives the net change
Interpreting definite integral as net change
Worked examples: interpreting definite integrals in context
Analyzing problems involving definite integrals
Quiz 1
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Start Quiz
Finding the area between curves expressed as functions of x
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Area between a curve and the x-axis
Area between a curve and the x-axis: negative area
Area between curves
Worked example: area between curves
Composite area between curves
Practice
Finding the area between curves expressed as functions of y
Learn
Area between a curve and the 𝘺-axis
Horizontal area between curves
Practice
Horizontal areas between curves
Not started
Finding the area between curves that intersect at more than two points
Learn
Practice
Area between curves that intersect at more than two points (calculator-active)
Not started
Quiz 2
Answer questions from recent lessons
Start Quiz
Quiz 3
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Start Quiz
Volumes with cross sections: squares and rectangles
Learn
Volume with cross sections: intro
Volume with cross sections: squares and rectangles (no graph)
Volume with cross sections perpendicular to y-axis
Practice
Volumes with cross sections: square and rectangles (intro)
Not started
Volumes with cross sections: triangles and semicircles
Learn
Volume with cross sections: semicircle
Volume with cross sections: triangle
Practice
Volumes with cross sections: triangles and semicircles
Not started
Harder volumes with cross sections
Learn
Volume of a pyramid through volume by slicing
Volume of a chopped out wedge
Practice
Volumes with disc method: revolving around x- or y-axis
Learn
Disc method around x-axis
Generalizing disc method around x-axis
Disc method around y-axis
Practice
Disc method: revolving around x- or y- axis
Not started
Volumes with disc method: revolving around other axes
Learn
Disc method rotation around horizontal line
Disc method rotating around vertical line
Calculating integral disc around vertical line
Practice
Disc method: revolving around other axes
Not started
Volume with washer method: revolving around x- or y- axis
Learn
Solid of revolution between two functions (leading up to the washer method)
Generalizing the washer method
Practice
Washer method: revolving around x- or y- axis
Not started
Volume with washer method: revolving around other axes
Learn
Washer method rotating around horizontal line (not x-axis), part 1
Washer method rotating around horizontal line (not x-axis), part 2
Washer method rotating around vertical line (not y-axis), part 1
Washer method rotating around vertical line (not y-axis), part 2
Practice
Washer method: revolving around other axes
Not started
Quiz 4
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Using integration to find the weighted average
Learn
Introduction to a weighted average using integration
Weighted average of a constant function
Weighted average worked example: cooking soup
Introduction to center of mass
Center of mass of a continuous object
Finding probability using integration
Learn
Introduction to probability using integration
Probability as a weighted average
Practice
Probability with integration
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Quiz 5
Answer questions from recent lessons
Start Quiz
Quiz 6
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The arc length of a smooth, planar curve and distance traveled
Learn
Arc length intro
Worked example: arc length
Practice
Arc length
Not started
Finding surface area using integration
Learn
Introduction to surface area using integration
Surface area of a sphere using integration
Setting up a definite integral for surface area pt. 1
Setting up a definite integral for surface area pt. 2
Surface area of a torus pt. 1
Quiz 7
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Calculator-active practice
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Practice
Contextual and analytical applications of integration (calc-active)
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Quiz 8
Answer questions from recent lessons
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Unit Test
Answer questions from the entire unit
Start Test
Worked example: motion problems (with definite integrals)
Average acceleration over interval
Practice
Analyzing motion problems (integral calculus)
Not started
Motion problems (with integrals)
Not started
Analyzing problems involving definite integrals
Worked example: problem involving definite integral (algebraic)
Practice
Interpreting definite integrals in context
Not started
Analyzing problems involving definite integrals
Not started
Problems involving definite integrals (algebraic)
Not started
Area between a curve and the x-axis
Not started
Area between two curves given end points
Not started
Area between two curves
Not started
Volumes with cross sections: squares and rectangles
Not started
Center of mass of a rode with variable density
Center of mass of a thin metal plate with variable density (pt. 1)
Center of mass of a thin metal plate with variable density (pt. 2)
Center of mass of a volume with variable density (pt. 1)
Center of mass of a volume with variable density (pt. 2)
Practice
Finding the weighted average
Not started
Surface area of a torus pt. 2
Practice
Surface area using integration
Not started