Study Notes: Unit 8 Area and Volume

Area Between Curves

The area enclosed between two or more curves, moving beyond just finding the area under a single curve.

Vertical Slicing Method

Integration method that sums vertical rectangles with heights determined by the difference of functions.

Area Formula for Vertical Slicing

A = ∫[f(x) - g(x)] dx, for continuous functions f and g where f(x) ≥ g(x).

Horizontal Slicing Method

Integration method that sums horizontal slices with heights derived from functions defined in terms of y.

Area Formula for Horizontal Slicing

A = ∫[f(y) - g(y)] dy, for continuous functions f and g with curves defined as functions of y.

Mnemonic for dx integration

Top minus Bottom for vertical slicing.

Mnemonic for dy integration

Right minus Left for horizontal slicing.

Volume of Solids with Cross Sections

Calculated by summing the volumes of infinitesimal slices, integrated over the base area.

Volume Formula

V = ∫A(x) dx or V = ∫A(y) dy, where A is the area of the cross section.

Cross-Section Shape: Square

Area formula is A(s) = s².

Cross-Section Shape: Semicircle

Area formula is A(s) = (π/8) * s².

Cross-Section Shape: Equilateral Triangle

Area formula is A(s) = (√3/4) * s².

Disc Method

Volume calculation method for solids of revolution where the region is flush against the axis.

Washer Method

Volume calculation method for solids of revolution with a gap, resulting in a hole in the middle.

Disc Volume Formula

V = π ∫[R(x)]² dx, where R(x) is the outer radius.

Washer Volume Formula

V = π ∫([R(x)]² - [r(x)]²) dx, where R(x) and r(x) are outer and inner radii.

Handling Rotations Around Shifted Axes

Adjust radius calculation by taking the absolute difference between the curve and axis.

Common Mistake: Algebra Error

Misapplying the integral for washers as (R - r)² instead of R² - r².

Common Mistake: Forgetting Pi

Calculating volume without including π in formulas involving circles.

Common Mistake: Mixing Variables

Using x-limit for a dy integral or vice versa.

Common Mistake: Determining Boundaries

Assuming integration bounds without finding intersection points.

Definition of R(x)

Outer radius in washer method, the distance from the axis to the far curve.

Definition of r(x)

Inner radius in washer method, the distance from the axis to the near curve.

Sketching the Region

First step in solving area problems; determine the top and bottom functions.

Finding Intersection Points

Step involving setting f(x) = g(x) to find boundaries of integration.

Setting Up the Integral

Applying Top - Bottom or Right - Left formulation in integral setup.

Interval Not Given

Set f(x) = g(x) to determine a and b for integration limits.