Applications of Integrals: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Limits & Continuity
• Derivatives
• Applications of Derivatives
• Integration
• Parametric & Polar

Area Between Curves

A = ∫[a,b] |f(x) - g(x)| dx. Find intersections; integrate difference.

Volume of Solids (Disk/Washer)

V = π∫[a,b] R²(x) dx (disk) or π∫[a,b] (R² - r²) dx (washer). Rotate curve around axis.

Volume of Solids (Shell Method)

V = 2π∫[a,b] x·f(x) dx. Integrate around vertical axis when washer method difficult.

Arc Length

L = ∫[a,b] √(1 + (f'(x))²) dx. Length of curve from a to b.

Surface Area

S = 2π∫[a,b] f(x)√(1 + (f'(x))²) dx. Area of surface of revolution.

Average Value

f_avg = (1/(b-a))∫[a,b] f(x) dx. Height of rectangle with same area as curve.

Accumulation Functions

F(x) = ∫[a,x] f(t) dt. F'(x) = f(x) (Fundamental Theorem). Accumulates area from a to x.

Differential Equations

dy/dx = f(x,y). Solving yields y as function of x. Example: exponential growth dy/dx = ky.

Separation of Variables

Rearrange: g(y) dy = f(x) dx. Integrate both sides. ∫g(y) dy = ∫f(x) dx + C.

Unit 5 Summary

Integrals calculate area, volume, arc length, surface area. Differential equations model real-world growth/decay.

Area Between Curves

∫(a to b) (top - bottom) dx, or (right - left) dy. Find every intersection first, because the roles can swap mid-interval.

Choosing dx or dy

Integrate in the variable that avoids splitting the region. Horizontal slices often collapse two dx-integrals into a single dy-integral.

Disk Method

V = π∫ R(x)² dx when the region touches the axis of revolution. R is the distance from the axis to the curve, not the y-value alone.

Washer Method

V = π∫ (R_outer² - R_inner²) dx. The squares are subtracted, never the radii first — (R-r)² is a different and wrong quantity.

Revolving About a Shifted Axis

About y = k, radii become |f(x) - k|. Forgetting the shift is the single most common volume error on the exam.

Known Cross Sections

V = ∫ A(x) dx where A is the area of the cross section: s² for squares, (√3/4)s² for equilateral triangles, (π/8)s² for semicircles on diameter s.

Arc Length in Cartesian Form

L = ∫(a to b) √(1 + (dy/dx)²) dx. The 1 comes from the horizontal run and is never omitted.

Arc Length Parametrically

L = ∫(t1 to t2) √((dx/dt)² + (dy/dt)²) dt, the parametric form that also underlies speed in motion problems.

Speed as an Integrand

For a particle in the plane, speed is √((dx/dt)² + (dy/dt)²), so distance traveled is the arc length of the path over that time interval.

Accumulation in Context

If R(t) is a rate in gallons per hour, ∫R dt gives gallons, and the initial amount must be added to obtain the total present at time t.

Two-Rate Problems

With an inflow rate I(t) and outflow O(t), the amount is A(0) + ∫(I - O)dt, and the maximum occurs where I = O with the difference switching from + to -.

Average Value in Applications

The average rate of flow over [a,b] is (1/(b-a))∫R dt — distinct from the average of the endpoint values unless R is linear.

Work Interpretation

Force integrated over displacement gives work; in BC contexts the same structure appears whenever a variable rate multiplies a small increment.

Improper Volumes

Revolving an unbounded region can give a finite volume: Gabriel's horn from y=1/x on [1,∞) has volume π but infinite surface area.

Region Bounded by Three Curves

When the upper boundary changes at an intersection, split the integral at that x-value and sum the pieces rather than forcing one formula.

Volume Setup Checklist

Identify the axis, decide slicing direction perpendicular to it, write radii as distances, and set limits from the intersection points.

Comparing Disks and Shells

Shells integrate 2π(radius)(height) in the variable parallel to the axis; either method works but one usually avoids splitting.

Signed vs Geometric Area

A definite integral counts area below the axis as negative; a geometric area question requires absolute values or splitting at the zeros.

Interpreting Units of an Integral

Units of ∫f(x)dx are the product of the units of f and of x — a check that catches setup errors quickly in applied problems.

Cross Sections Perpendicular to the y-Axis

When slices are perpendicular to the y-axis, express the side length in terms of y and integrate dy between the y-limits.