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What is the formula for the average value of a continuous function f over an interval [a,b]
The average value of a continuous function f over an interval [a,b] is given by: 1/b−a∫abf(x) dx
What does the definite integral of speed represent for a particle in rectilinear motion over an interval of time?
The definite integral of speed represents the particle's total distance traveled over the interval of time.
What does a function defined as an integral represent?
A function defined as an integral represents an accumulation of a rate of change.
What does the definite integral of the rate of change of a quantity over an interval give?
The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval
How can the definite integral be used in applied contexts?
The definite integral can be used to express information about accumulation and net change in many applied contexts.
How can areas of regions in the plane be calculated?
Areas of regions in the plane can be calculated with definite integrals.
Can areas of regions in the plane be calculated using functions of either x or y?
Yes, areas of regions in the plane can be calculated using functions of either xxx or yyy.
How can areas of certain regions in the plane be calculated?
Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.
How can volumes of solids with square and rectangular cross sections be found?
Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.
How can volumes of solids with triangular cross sections be found?
Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes.
How can volumes of solids with semicircular and other geometrically defined cross sections be found?
Volumes of solids with semicircular and other geometrically defined cross-sections can be found using definite integrals and the area formulas for these shapes.
How can volumes of solids of revolution around the x- or y-axis be found?
Volumes of solids of revolution around the x- or y-axis may be found by using definite integrals with the disc method.
How can volumes of solids of revolution around any horizontal or vertical line in the plane be found
Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.
How can volumes of solids of revolution around the x- or y-axis whose cross sections are ring shaped be found?
Volumes of solids of revolution around the x- or y-axis whose cross sections are ring- shaped may be found using definite integrals with the washer method.
How can volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped be found?
Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.
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