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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.
Introduces the disk method with worked examples of finding the volume of a right circular cone and a sphere.
Generalization of the disk method when cross-sectional areas are known. Includes worked example.
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.
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.
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).
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.
Introduces surface area of a surface of revolution using the case of a right circular cone.
Formula for the surface area of a surface of revolution. Includes examples of a line segment, a semicircle, and an astroid.
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.
Method for finding the area of a region bounded by a polar curve, using the example of a cardioid.
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.
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.
Definition as a method for finding the area of a volume under a surface defined by a function of x and y.
Proving the volume of a cone using the washer method for finding volumes of solids of revolution.
Sketching and computing the area of the polar curve r = cos(3*θ).
Finding the volume of a solid of revolution about the x-axis.
Finding the volume of a solid of revolution about the y-axis.
Eighteen problems with answers but not complete solutions on these four topics.
Finding the volume of a solid of revolution about the x-axis.
Finding the area of the region in the 1st and 3rd quadrants between two circles defined in polar coordinates.
Two problems which involve finding the area between a curve and a tangent line and the area between two parabolas.
Finding the volume of a solid of revolution about the x-axis.
Finding the arc length of a segment of the graph of the natural logarithm.
Setting up a definite integral for the amount of money in an account at the end of a year.
Finding the volume of a glass vase in the shape of a solid of revolution
Finding the volume of ice cream in an overfilled cone defined by a solid of revolution.
Finding the average value of rectangles inscribed at random in a quarter of a circle.
Finding the volume of candy needed to fill a Great Pumpkin with shape defined in terms of a solid of revolution.
Finding the average area of slices of a SmartHam defined in terms of a solid of revolution.
Finding the volume of a solid of revolution about the y-axis.
Finding the average amount score and average amount of sleep students get the night before a test.
Finding the volume of a solid of revolution about the y-axis.
Sketching a curve defined in polar coordinates and finding the area inside it.
Finding the length of a curve defined parametrically.
Setting up an integral for the length of one arc of the sine curve, and estimating the value of the integral.
Setting up and evaluating an integral for the mass of a disc with variable mass density.
Finding the volume of a solid of revolution about the y-axis.
Setting up and estimating the value of an integral representing the length of a given curve.
Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.
Setting up an integral to find the length of a curve given in rectangular coordinates.
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.
Finding the amount of wine that can be held in a glass defined in terms of a solid of revolution about the y-axis.
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.
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.