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Area Between Curves

Learn area between curves, area with parallel boundaries.

Lesson 1 of 32 parts Practice Flashcards

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1

Area Between Curves

Objective: Explain area between curves, apply the stated method, and check its conditions.

Where upper curve f and lower curve g maintain order, area is ∫(f−g)dx. If they cross, split at intersection inputs so each integrand uses the actual upper curve. Area is nonnegative and uses square units when both axes measure lengths. Split the interval if the curves switch vertical order.

Formulas

$Area=∫(upper−lower)dx$

Worked examples

Try itFind area between y=6x and y=0 on [0,7].Show answer

Step 1: The upper curve is ax on this interval. Step 2: Integrate ax−0 from 0 to 7. Step 3: Area is 147 square units.

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1

Area Between Curves

Objective: Explain area between curves, apply the stated method, and check its conditions.

Where upper curve f and lower curve g maintain order, area is ∫(f−g)dx. If they cross, split at intersection inputs so each integrand uses the actual upper curve. Area is nonnegative and uses square units when both axes measure lengths. Split the interval if the curves switch vertical order.

Formulas

$Area=∫(upper−lower)dx$

Worked examples

Try itFind area between y=6x and y=0 on [0,7].Show answer

Step 1: The upper curve is ax on this interval. Step 2: Integrate ax−0 from 0 to 7. Step 3: Area is 147 square units.

2

Area with Parallel Boundaries

Objective: Explain area with parallel boundaries, apply the stated method, and check its conditions.

Between two horizontal curves y=M and y=m with M≥m on an interval of width w, area is (M−m)w. This is a simple case of upper-minus-lower integration. Neither boundary’s absolute height alone determines the enclosed area. Use the difference between boundaries.

Formulas

$Area=(M−m)(b−a)$

Worked examples

Try itFind area between y=13 and y=7 on [0,7].Show answer

Step 1: Upper minus lower is 13−7=6. Step 2: Integrate the constant 6 over width 7. Step 3: Area is 42.

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