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Formula for the average value of a function f(x) on [a, b]?

$\frac{1}{b-a} \int_{a}^{b} f(x) , dx$

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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?

$\int (1/2)[object Object](f(x))^2*w , dx$

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]?

$\int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx$

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.

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].

All Flashcards

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?

$\int (1/2)[object Object](f(x))^2*w , dx$

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]?

$\int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx$

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.

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].