Calculus I · Unit 3B · lesson

Area Between Curves

Concept

Learning objectives

Set up and evaluate area integrals using top-minus-bottom or right-minus-left.

Area Between Curves

Explanation

Area is built from the length of a representative slice

For a vertical slice, the local height of a region is top minus bottom. Multiplying that height by a small width $dx$ gives a thin rectangle whose area approximates the corresponding piece of the region. Integrating adds these pieces. The formula is therefore not a slogan to memorize but a direct description of the geometry of one slice.

The first task is to find intersections and determine which curve is on top over each interval. If the order changes, the setup must be split. A negative result is a warning that the curves were subtracted in the wrong order, because geometric area cannot be negative. Sketching the region and drawing one labeled slice is usually faster than repairing an integral assembled blindly.

For vertical slices,

A=\int_a^b[\text{top}-\text{bottom}]dx.

For horizontal slices,

A=\int_c^d[\text{right}-\text{left}]dy.

Guided walkthrough

Area enclosed by a line and parabola

Find the area between $y=2x$ and $y=x^2$ .

Answer reveal

Worked solution

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Interactive checku3b-area-01

Find the area between $y=2x$ and $y=x^2$ from $x=0$ to $x=2$ .

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Show hint

Integrate top minus bottom.

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**Visual study**
Vertical slices between two curves. Show top-minus-bottom slice height and accumulated region.

After the explanation

Use the section idea

Reading lens

Let the axis and slice orientation determine every distance; top-minus-bottom, right-minus-left, radii, and shell height must all come from the same picture.

Mental model

Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.

Decision

Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.

Common trap

Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.

Check yourself

Do the slice dimensions remain nonnegative on the full interval, and do their units multiply to area or volume?

Source & rights

Original instruction with traceable references.

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