Revise later
SpaceTo flip
If confident
All Flashcards
Define the average value of a function.
The value a function would take at a single point if the area under the curve equaled the area of a rectangle with the same width and height.
What is displacement?
A vector quantity representing the change in position of an object.
Define total distance traveled.
A scalar quantity representing the total distance covered by an object, regardless of its final position.
What is a solid of revolution?
A three-dimensional shape formed by rotating a two-dimensional region about an axis.
Define arc length.
A measure of the distance along the curved path of a function.
What is velocity?
The rate of change (derivative) of the position as a function of time.
What is acceleration?
The rate of change (derivative) of velocity as a function of time.
Define the disc method.
A method to find the volume of a solid of revolution by cutting it into thin disks.
Define the washer method.
A method to find the volume of a solid of revolution using ring-shaped objects (washers).
What is a cross-section?
The intersection of a solid with a plane.
Formula for the average value of a function f(x) on [a, b]?
$ \frac{1}{b-a} \int_{a}^{b} f(x) , dx $
Formula for displacement given velocity v(t) on [a, b]?
$ \int_{a}^{b} v(t) , dt $
Formula for total distance traveled given velocity v(t) on [a, b]?
$ \int_{a}^{b} |v(t)| , dt $
Formula for area between curves f(x) and g(x) on [a, b], where f(x) > g(x)?
$ \int_{a}^{b} [f(x) - g(x)] , dx $
Formula for volume with square cross-sections?
$ \int [f(x)]^2 , dx $
Formula for volume with rectangular cross-sections?
$ \int f(x) * w , dx $
Formula for volume with triangular cross-sections?
$ \int (1/2)*f(x)*w , dx $
Formula for volume with semicircular cross-sections?
$ \int (1/2)*\pi*(f(x))^2*w , dx $
Formula for volume using the disc method?
$ \int \pi [f(x)]^2 , dx $
Formula for volume using the washer method?
$ \int \pi (R^2 - r^2) , dx $
Formula for arc length of a curve y = f(x) on [a, b]?
$ \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx $
Difference between displacement and total distance traveled?
Displacement: Change in position, can be negative. | Total Distance: Total path length, always non-negative.
Difference between disc and washer method?
Disc: Solid of revolution has no hole. | Washer: Solid of revolution has a hole.
Difference between integrating with respect to x and with respect to y when finding areas?
x: Use vertical rectangles, integrate along x-axis. | y: Use horizontal rectangles, integrate along y-axis.
Difference between finding area between curves using functions of x vs functions of y?
Functions of x: Integrate (top - bottom) with respect to x. | Functions of y: Integrate (right - left) with respect to y.
Compare finding volumes by slicing vs. using disc/washer method.
Slicing: General method for any cross-sectional shape. | Disc/Washer: Specific to solids of revolution with circular/ring cross-sections.
Compare the use of definite integrals in finding area and volume.
Area: Integrates a function representing height to find 2D space. | Volume: Integrates a function representing area to find 3D space.
Compare using cross-sections of squares vs. circles for volume calculation.
Squares: Volume is integral of (side length)^2. | Circles: Volume is integral of $ \pi * (radius)^2 $.
Compare the impact of axis of rotation on disc vs. washer method.
Disc: Radius is the distance from the function to the axis. | Washer: Requires both inner and outer radii relative to the axis.
Compare the use of integrals in finding net change vs. total accumulation.
Net Change: Direct integration gives the difference between endpoints. | Total Accumulation: Requires considering absolute values or intervals of increase/decrease.
Compare the application of integrals in physics vs. economics.
Physics: Used for motion, work, energy. | Economics: Used for cost, revenue, consumer surplus.