10 Applications of Integration Free Online FlashCards

What modeling principle underlies applications of definite integrals?

A definite integral accumulates infinitely many small contributions: total quantity = ∫(quantity per unit of the variable) d(variable).

How is area between curves computed with vertical slices?

For vertical slices, area is ∫_a^b [f(x) − g(x)] dx, where f is the top curve and g is the bottom curve.

What should you do when two curves cross?

Find the intersection points, split the interval there, and add the positive areas using top minus bottom on each subinterval.

What is the horizontal-slice formula for area?

For horizontal slices, area is ∫_c^d [R(y) − L(y)] dy, where R(y) is the right boundary and L(y) is the left boundary.

What is the general slicing formula for volume?

If A(x) is the cross-sectional area perpendicular to the x-axis, then the solid’s volume is V = ∫_a^b A(x) dx.

What is the disk-method formula for rotation about the x-axis?

For rotation about the x-axis, V = π∫_a^b [f(x)]² dx, because each cross section is a disk of radius f(x).

What formula gives the volume of a washer-shaped solid?

For outer radius R(x) and inner radius r(x), V = π∫_a^b ([R(x)]² − [r(x)]²) dx.

What is the cylindrical-shell formula about the y-axis?

For rotation about the y-axis, V = 2π∫_a^b x f(x) dx: radius x, height f(x), and thickness dx.

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

The average value is f_avg = 1/(b − a) ∫_a^b f(x) dx.

How is the mass of a rod with variable linear density found?

For linear density ρ(x), mass is m = ∫_a^b ρ(x) dx. The density measures mass per unit length.

What formula gives the mass of a disk with radial density?

For radial density ρ(r) on a disk of radius R, m = ∫_0^R 2πrρ(r) dr.

How is work calculated for a variable force?

Work against a variable force F(x) over [a,b] is W = ∫_a^b F(x) dx.