Appropriate Integrals

 

Areas Between Curves

Method for finding the area between two curves. Includes worked example of finding the area bounded by the curve y=x(x2-3) and a horizontal tangent line.

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Volumes of Solids of Revolution: The Disk Method

Introduces the disk method with worked examples of finding the volume of a right circular cone and a sphere.

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The Slice Method

Generalization of the disk method when cross-sectional areas are known. Includes worked example.

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Volumes of Solids of Revolution: The Washer Method

Variation of disk method using the difference of two disks to create washers. Includes worked example of finding the volume of material of a dog dish.

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Average Value of a Function

Derivation of average value formula using Reimann sums. Example of finding the average radius r(x) = r0 + Acos(wx) of a wire made by a vibrating machine.

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Finding the average value of rectangles inscribed at random in a quarter of a circle.

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Volumes of Solids of Revolution: The Shell Method

Explanation of the shell method as an alternative to the disk and washer methods. Revisits worked example of finding the volume of material of a dog dish (previously solved using the washer method in section 5 of lecture 19).

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Arc Length

Derivation of formula for finding the length of a curve. Includes worked examples. Example 2 demonstrates finding the length of a curve with equation y = f(x) by changing to parametric equations.

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Surface Area of a Right Circular Cone

Introduces surface area of a surface of revolution using the case of a right circular cone.

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Surface Area of a Surface of Revolution

Formula for the surface area of a surface of revolution. Includes examples of a line segment, a semicircle, and an astroid.

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Arc Length in Polar Coordinates

Method for finding arc length of a polar curve, including example of a cardioid. Method for finding surface area of a surface of revolution, with example of a cardioid used to approximate the surface area of an apple.

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Area of a Region Enclosed by a Polar Curve

Method for finding the area of a region bounded by a polar curve, using the example of a cardioid.

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Review Problems

Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.

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Average Value

Finding the average value of a function over an interval, with diagrams and examples relating to temperature, alternating current, and chords in a unit circle.

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Area Integrals

Definition as a method for finding the area of a volume under a surface defined by a function of x and y.

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Washer Method

Proving the volume of a cone using the washer method for finding volumes of solids of revolution.

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Three-Leaved Rose

Sketching and computing the area of the polar curve r = cos(3*θ).

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Solids of Revolution: X-Axis

Finding the volume of a solid of revolution about the x-axis.

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Solids of Revolution: Y-Axis

Finding the volume of a solid of revolution about the y-axis.

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Miscellaneous Integration Problems

Eighteen problems with answers but not complete solutions on these four topics.

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Solids of Revolution

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Finding the volume of a solid of revolution about the x-axis.

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Finding the volume of a solid of revolution about the y-axis.

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Area of a Lune

Finding the area of the region in the 1st and 3rd quadrants between two circles defined in polar coordinates.

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Area Between Two Curves

Two problems which involve finding the area between a curve and a tangent line and the area between two parabolas.

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Volume of Solid of Revolution

Finding the volume of a solid of revolution about the x-axis.

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Arc Length of a Segment of a Curve

Finding the arc length of a segment of the graph of the natural logarithm.

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Continuously Compounded Interest

Setting up a definite integral for the amount of money in an account at the end of a year.

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Solids of Revolution: Volume of A Vase

Finding the volume of a glass vase in the shape of a solid of revolution

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Solids of Revolution: Ice Cream Cone

Finding the volume of ice cream in an overfilled cone defined by a solid of revolution.

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Solids of Revolution: Great Pumpkin

Finding the volume of candy needed to fill a Great Pumpkin with shape defined in terms of a solid of revolution.

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Average Value: SmartHam

Finding the average area of slices of a SmartHam defined in terms of a solid of revolution.

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Average Value: Test Scores

Finding the average amount score and average amount of sleep students get the night before a test.

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Area Inside a Polar Curve

Sketching a curve defined in polar coordinates and finding the area inside it.

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Arc Length of a Curve

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Finding the length of a curve defined parametrically.

Setting up and estimating the value of an integral representing the length of a given curve.

Setting up an integral to find the length of a curve given in rectangular coordinates.

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Estimating the Length of the Sine Curve

Setting up an integral for the length of one arc of the sine curve, and estimating the value of the integral.

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Mass of a Metal Disc

Setting up and evaluating an integral for the mass of a disc with variable mass density.

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Integrals in Polar Coordinates: Lunar Eclipse

Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.

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Polar Coordinates: Spiral

Sketching a spiral defined in polar coordinates, counting the times it crosses the x-axis, and finding the area of specific regions of the spiral.

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Solids of Revolution: Wine Glass

Finding the amount of wine that can be held in a glass defined in terms of a solid of revolution about the y-axis.

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Average Value: Suspension Bridge

Setting up an integral for the length of the main cables in a suspension bridge, and using it to find the average length of the vertical cables connecting to the roadway.

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Polar Curves

Sketching a curve given in polar coordinates and finding the area swept by a line segment as one of the endpoints moves along this curve.

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