Applications of Integration Free Online FlashCards

What is the central modeling idea behind applications of integration?

Total amount = ∫ rate or density d(independent variable). Integration adds infinitely many small contributions such as slices, rectangles, or shells.

How do you find area between curves using vertical slices?

A = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) is the top curve and g(x) is the bottom curve. Split the interval if the curves cross.

How do you find area between curves using horizontal slices?

A = ∫𝑐ᵈ [R(y) − L(y)] dy, where R(y) is the right boundary and L(y) is the left boundary.

What is the general formula for volume by slicing?

V = ∫ₐᵇ A(x) dx, where A(x) is the cross-sectional area perpendicular to the axis of integration.

What formula gives volume by washers?

V = π∫ₐᵇ(R² − r²) dx. R is the outer radius and r is the inner radius, both measured as distances from the axis of rotation.

What formula gives cylindrical-shell volume about the y-axis?

V = 2π∫ₐᵇ x[f(x) − g(x)] dx when rotating a region about the y-axis. More generally, integrate circumference × height with respect to radius.

How should you choose between washers and shells?

Choose the method that gives the simplest radius, height, and limits. Washers use slices perpendicular to the axis; shells use slices parallel to the axis.

What is the average value of f on [a,b]?

f_avg = (1/(b − a))∫ₐᵇ f(x) dx. This averages output values; average rate of change is [f(b) − f(a)]/(b − a).

What is the derivative of an accumulation function?

If A(x) = ∫ₐˣ r(t) dt, then A′(x) = r(x). The derivative of the accumulation function recovers its rate.

How do you reconstruct a quantity from its initial value and rate?

Q(t) = Q₀ + ∫ₐᵗ r(u) du. Final amount equals initial amount plus accumulated net change.

What does the integral of velocity represent?

Displacement is the net change in position: s(b) − s(a) = ∫ₐᵇ v(t) dt. It may be positive, negative, or zero.

How is total distance found from velocity?

Distance traveled = ∫ₐᵇ |v(t)| dt. Find where v(t) = 0 and split the interval so negative velocity does not cancel positive velocity.