Applications of Integration

Thirteen topics, one idea: integrate a cross-section. The average value of a function is an integral divided by the interval; position needs the starting value added and total distance needs the absolute value; the area between curves is top minus bottom, with a split at every crossing; a cross section contributes its own area, so a square gives s² and a washer gives R² − r², never (R − r)²; and a radius is always measured from the axis you are revolving about.

AB exam 10-15%BC exam 5-10%13 topics

Topics

8.1

Finding the Average Value of a Function on an Interval

Connecting Position, Velocity, and Acceleration of Functions Using Integrals

Using Accumulation Functions and Definite Integrals in Applied Contexts

Finding the Area Between Curves Expressed as Functions of x

Finding the Area Between Curves Expressed as Functions of y

Finding the Area Between Curves That Intersect at More Than Two Points

Volumes with Cross Sections: Squares and Rectangles

Volumes with Cross Sections: Triangles and Semicircles

Volume with Disc Method: Revolving Around the x- or y-Axis

Volume with Disc Method: Revolving Around Other Axes

Volume with Washer Method: Revolving Around the x- or y-Axis

Volume with Washer Method: Revolving Around Other Axes

The Arc Length of a Smooth, Planar Curve and Distance TraveledBC only

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Average value

the integral DIVIDED by b − a. A value, not an integral

Divide by what

b − a, the LENGTH: never b, never the sample count

MVT for integrals

a continuous f on [a, b] ATTAINS its average at some c in [a, b]

Net change

the integral of a RATE is the CHANGE, not the amount

Final amount

starting value PLUS the integral. The constant is not optional

Displacement

integrate v itself; sign and all

Total distance

integrate |v|, so SPLIT wherever v = 0

Speed

|v|. It rises when v and a share a sign

Units

rate units TIMES the units of the variable integrated

Saying what it means

name the quantity, the interval, and the units

With respect to x

TOP minus BOTTOM, dx, limits are x-values

With respect to y

RIGHT minus LEFT, dy, limits are y-values

Which one

pick the variable that needs FEWER splits

Going to dy

solve for x FIRST; y-limits, not the old x-limits

Which is on top

TEST a point between the crossings. Do not eyeball it

Negative answer

the order was reversed. Area is never negative

The limits

solve f = g for ALL roots; given numbers are not bounds

Curves that cross

one integral per piece, each ordered on its own

Why splitting matters

one integral across a crossing CANCELS, it does not add

Absolute value

|f − g| says split; it is not a repair applied at the end

Every volume

integrate ONE SLICE'S AREA along the perpendicular axis

The side s

the DISTANCE between the boundaries: top minus bottom

Square section

s²

Rectangle

s times the stated multiple of s

Isosceles right △

s²/2 with s a LEG; s²/4 with s the hypotenuse

Equilateral △

(√3/4)s², not (√3/2)s²

Semicircle

(π/2)r² with r = s/2, so (π/8)s²

Disc

π times radius squared, integrated

Washer

π(R² − r²). NEVER π(R − r)²

Radius

a DISTANCE from the axis, so it is never negative

About y = k

|f(x) − k|, adjusted on BOTH radii

About x = k

solve for x, integrate dy, radius |g(y) − k|

Arc length (BC)

√(1 + (f′)²) dx. The 1, the square, the root

Distance travelled (BC)

the same integral: arc length of the path

Unit 8 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 13 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from. AB students are served only the AB items; BC students get the whole unit.

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Check what stuck.

Units 1 through 8, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

20 or 45questions

81topics

116codes covered

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