Area under a curve and the meaning of a definite integral

Key idea

The definite integral \(\displaystyle \int_a^b f(x)\,dx\) measures net accumulation over \([a,b]\). Geometrically, it is the signed area between the graph of \(y=f(x)\) and the \(x\)-axis:

• Area above the \(x\)-axis counts as positive.
• Area below the \(x\)-axis counts as negative.

If \(f(x)\ge 0\) on \([a,b]\), then the integral equals the usual (positive) area: \[ A=\int_a^b f(x)\,dx. \] If the graph crosses the axis, and you want total area, you break the interval at the zeros of \(f\) and add the absolute areas.

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Summary

• \(\int_a^b f(x)\,dx\) is net (signed) area/accumulation over \([a,b]\).
• If \(f(x)\ge 0\), it equals the geometric area under the curve.

Area between curves: top minus bottom, with correct bounds

Key idea

To find the area enclosed between two curves \(y=f(x)\) and \(y=g(x)\) over \([a,b]\), sketch or compare values to decide which is on top. Then use: \[ A=\int_a^b \bigl(\text{top} - \text{bottom}\bigr)\,dx. \] The hardest part is usually picking the correct bounds \(a\) and \(b\). You typically find them by solving \(f(x)=g(x)\).

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Summary

• Area between curves uses \(\int(\text{top}-\text{bottom})\,dx\) over the correct intersection bounds.
• Sketching (even a quick one) helps you choose top/bottom and avoid sign mistakes.

Volumes of revolution: disk method and washer method

Key idea

When a region is rotated about an axis, the cross-sections perpendicular to that axis are circles. If you rotate a region about the \(x\)-axis, the radius is a vertical distance from the curve to the axis:

• Disk method (no hole): \(\displaystyle V=\pi\int_a^b [R(x)]^2\,dx\)
• Washer method (hole): \(\displaystyle V=\pi\int_a^b\big([R(x)]^2-[r(x)]^2\big)\,dx\)

A reliable checklist:

• Sketch the region and axis of rotation.
• Write \(R(x)\) (outer radius) and \(r(x)\) (inner radius) as distances to the axis.
• Square the radii, subtract if needed, multiply by \(\pi\), integrate over the correct bounds.

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Summary

• Disk: \(V=\pi\int R^2\). Washer: \(V=\pi\int(R^2-r^2)\).
• Always define radii as distances to the axis of rotation.

Volume with cylindrical shells: \(2\pi\int (\text{radius})(\text{height})\)

Key idea

The shell method uses cylindrical shells instead of disks. A typical setup (rotating around the \(y\)-axis with vertical slices) is: \[ V=2\pi\int_a^b (\text{radius})(\text{height})\,dx. \] For shells around the \(y\)-axis:

• radius \(=x\) (distance from slice at \(x\) to the \(y\)-axis),
• height \(=\text{top}-\text{bottom}\) (vertical length of the region at that \(x\)).

Shells are often best when rotating around the \(y\)-axis but your functions are written as \(y=f(x)\), because washers may require rewriting in terms of \(x\) as a function of \(y\).

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Summary

• Shells: \(V=2\pi\int (\text{radius})(\text{height})\).
• Pick shells when they let you keep the problem in \(x\) (or in \(y\)) without solving for the other variable.

Surface area of a surface of revolution

Key idea

When you rotate a curve \(y=f(x)\) about the \(x\)-axis, you generate a surface. The surface area is found by combining circumference \(2\pi f(x)\) with an arc-length factor: \[ S=2\pi\int_{a}^{b} f(x)\sqrt{1+(f'(x))^2}\,dx, \] (assuming \(f(x)\ge 0\) on \([a,b]\)).

For rotation about the \(y\)-axis, a common form is: \[ S=2\pi\int_{a}^{b} x\sqrt{1+(f'(x))^2}\,dx, \] when rotating \(y=f(x)\) around the \(y\)-axis.

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Summary

• Surface area uses circumference \(\times\) arc length: \(S=2\pi\int f(x)\sqrt{1+(f'(x))^2}\,dx\).
• The square-root factor comes from converting \(dx\) into a small curve length \(ds\).

Common mistakes: wrong bounds, wrong “top,” and wrong radius

Key idea

Most errors in applications of integrals are not algebra errors — they are setup errors. Use this checklist before you integrate:

• Draw a quick sketch. Identify the region and axis of rotation.
• Find bounds. Use the interval given, or solve intersections like \(f(x)=g(x)\).
• Choose a method. Disk/washer for perpendicular slices; shells for parallel slices.
• Write the radius/height as distances. If rotating about \(y=c\), radius is \(|f(x)-c|\).
• Check units. Area \(\to\) squared units, volume \(\to\) cubic units.

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Summary

• Most mistakes are setup mistakes: bounds, top/bottom, and radius definitions.
• Use a sketch + distance-based radius/height to make your integrals correct.

Why applications of integrals matter

Where applications of integrals show up

• Geometry: exact areas, volumes, and surface areas that are hard to get by formulas alone.
• Physics: work, fluid pressure, mass with variable density, and centers of mass.
• Engineering: volumes and surface areas for design and material usage.
• Economics: accumulated change and total value from marginal functions.

Worked example: a clean “pattern” you can reuse

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Final recap

• Area under a curve: \(A=\int_a^b f(x)\,dx\) when \(f(x)\ge 0\).
• Area between curves: \(A=\int_a^b(\text{top}-\text{bottom})\,dx\).
• Disk/washer volume: \(V=\pi\int R^2\,dx\) or \(V=\pi\int(R^2-r^2)\,dx\).
• Shell volume: \(V=2\pi\int (\text{radius})(\text{height})\,dx\) (or \(dy\)).
• Surface area: \(S=2\pi\int f(x)\sqrt{1+(f'(x))^2}\,dx\) (about the \(x\)-axis).
• Setup first: sketch, bounds, distances, then integrate.