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Explain how definite integrals relate position, velocity, and acceleration.
The definite integral of velocity gives displacement, and the definite integral of acceleration gives velocity change.
Explain how to find the area between two curves.
Integrate the difference between the two functions over the interval of interest. Ensure you subtract the lower function from the upper function.
Explain the concept of accumulation functions.
Accumulation functions represent the accumulated change of a quantity over an interval, calculated using definite integrals.
Explain the disc method for finding volumes.
Divide the solid into thin disks, find the volume of each disk using $ \pi r^2 dx $, and integrate to find the total volume.
Explain the washer method for finding volumes.
Divide the solid into thin washers, find the area of each washer using $ \pi (R^2 - r^2) dx $, and integrate to find the total volume.
Explain the relationship between displacement and total distance traveled.
Displacement considers the change in position, while total distance traveled considers the entire path taken, regardless of direction.
How do you handle areas between curves that intersect multiple times?
Divide the interval into subintervals based on intersection points, integrate separately over each subinterval, and sum the absolute values of the results.
Explain the concept of finding volumes with cross-sections.
Determine the area of a representative cross-section, express it as a function of x or y, and integrate over the appropriate interval.
How does the choice of axis of rotation affect volume calculations?
The axis of rotation determines whether you integrate with respect to x or y and affects the setup of the radius in disc/washer method problems.
Explain how to find the arc length of a curve.
Use the formula $ \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx $ to calculate the length of the curve over the interval [a, b].
Steps to find the average value of f(x) on [a, b]?
1. Find the definite integral of f(x) from a to b. 2. Divide the result by (b-a).
Steps to find displacement given v(t) on [a, b]?
1. Integrate v(t) from a to b.
Steps to find total distance traveled given v(t) on [a, b]?
1. Find intervals where v(t) is positive and negative. 2. Integrate |v(t)| over [a, b], summing the absolute values of the integrals over each interval.
Steps to find the area between f(x) and g(x) on [a, b]?
1. Determine which function is greater. 2. Integrate the difference between the greater and lesser function from a to b.
Steps to find volume using the disc method?
1. Identify the radius f(x). 2. Integrate $ \pi [f(x)]^2 $ over the interval.
Steps to find volume using the washer method?
1. Identify the outer radius R and inner radius r. 2. Integrate $ \pi (R^2 - r^2) $ over the interval.
Steps to find volume with square cross-sections?
1. Determine the side length of the square, f(x). 2. Integrate $ [f(x)]^2 $ over the interval.
Steps to find the arc length of a curve y = f(x) on [a, b]?
1. Find the derivative f'(x). 2. Square the derivative. 3. Add 1. 4. Take the square root. 5. Integrate from a to b.
Steps to find the area between curves that intersect at multiple points?
1. Find the points of intersection. 2. Divide the interval into subintervals. 3. Integrate the absolute value of the difference between the functions over each subinterval. 4. Sum the results.
Steps to find volume of a solid with cross-sections perpendicular to the x-axis?
1. Express the area of the cross-section as a function of x, A(x). 2. Integrate A(x) with respect to x over the interval [a, b].
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 $