Source: Midterm 2 Review Sheet, Math 15200 Section 42
Tags: area between curves, integration with respect to x, integration with respect to y, disk method, washer method, shell method, volume of revolution, solid of revolution, calculus II, MATH 152
These sections cover two core skills: finding the area of a region trapped between curves, and finding the volume of solids formed by spinning those regions around an axis. You choose your method (disk/washer or shell) based on whether the axis of revolution matches your variable of integration.
Area between curves (with respect to x)
The integral of [top function minus bottom function] over a horizontal interval [a, b]. You integrate with respect to x when vertical slices cleanly capture the region.
Area between curves (with respect to y)
The integral of [right function minus left function] over a vertical interval [c, d]. You integrate with respect to y when horizontal slices are simpler.
Disk method
A special case of the washer method where the inner radius is zero, i.e. the region touches the axis of revolution. Produces a solid with no hole through the middle.
Washer method
Used to find volumes of revolution when the cross-section perpendicular to the axis is a ring (an outer circle minus an inner circle). The formula squares both the outer and inner radius functions and subtracts them under the integral.
Shell method
Computes volumes of revolution by summing up thin cylindrical shells rather than flat disks. Each shell has radius equal to the distance from the axis and height equal to the function value.
Axis of revolution
The line about which the region is rotated. Most commonly the x-axis or y-axis, but can be any horizontal or vertical line.
Integrating with respect to x
Region Ω sits between x = a and x = b, with y = f(x) on top and y = g(x) on the bottom.
Always subtract bottom from top: the integrand is f(x) − g(x).
If the curves cross within [a, b], split the integral at each crossing point and check which function is on top in each sub-interval.
Integrating with respect to y
Region Ω sits between y = c and y = d, with x = F(y) on the right and x = G(y) on the left.
Always subtract left from right: the integrand is F(y) − G(y).
This approach is often simpler when the boundary curves are easier to express as functions of y, or when integrating with respect to x would force you to split into multiple integrals.
Revolving around the x-axis (integrate with respect to x)
Cross-sections perpendicular to the x-axis are washers.
Outer radius = f(x), inner radius = g(x).
If the inner radius is zero (the region touches the axis), this simplifies to the disk method.
Revolving around the y-axis (integrate with respect to y)
Cross-sections perpendicular to the y-axis are washers.
Outer radius = F(y), inner radius = G(y).
When to use it: the disk/washer method works when the axis of revolution is the same as the variable of integration. Revolving around the x-axis and integrating dx, or revolving around the y-axis and integrating dy.
Revolving around the y-axis (integrate with respect to x)
Each shell has radius x (its distance from the y-axis) and height f(x) − g(x).
You integrate with respect to x even though the revolution is about the y-axis.
Revolving around the x-axis (integrate with respect to y)
Each shell has radius y (its distance from the x-axis) and height F(y) − G(y).
You integrate with respect to y even though the revolution is about the x-axis.
When to use it: the shell method works when the axis of revolution is different from the variable of integration. Revolving around the y-axis but integrating dx, or revolving around the x-axis but integrating dy.
Revolving about lines other than the axes
When the axis of revolution is a line like x = k or y = k (not the coordinate axes themselves), the radius expressions change.
For washers, the outer and inner radii become the distances from the curves to that line, not to the origin.
For shells, the shell radius becomes |x − k| or |y − k| depending on orientation.
This is a common source of exam errors. Always sketch the region and label the distances.
Area (with respect to x): A = ∫ from a to b of [f(x) − g(x)] dx
Area (with respect to y): A = ∫ from c to d of [F(y) − G(y)] dy
Disk/Washer volume (revolve about x-axis, integrate dx): V = ∫ from a to b of π · [f(x)]² − [g(x)]² dx
Disk/Washer volume (revolve about y-axis, integrate dy): V = ∫ from c to d of π · [F(y)]² − [G(y)]² dy
Shell volume (revolve about y-axis, integrate dx): V = ∫ from a to b of 2πx · [f(x) − g(x)] dx
Shell volume (revolve about x-axis, integrate dy): V = ∫ from c to d of 2πy · [F(y) − G(y)] dy
⚠️ The single most important decision on any volume problem is choosing between disk/washer and shell. The rule: if the axis of revolution matches your integration variable, use disk/washer. If they differ, use shell.
⚠️ Forgetting to square the radius functions in the washer formula is a very common mistake. The formula uses π(R² − r²), not π(R − r)².
⚠️ When curves cross within the interval, you must split the integral. The "top" and "bottom" functions can swap, and using the wrong one gives a negative area or wrong volume.
⚠️ Revolution about a line other than the axes changes every radius expression. If the problem says "revolve about x = 3" or "revolve about y = −1," rewrite the radius as the distance from the curve to that line before setting up the integral.
⚠️ For shell method, the factor out front is 2π (not π). For washer method, it is π (not 2π). Mixing these up is easy under time pressure.
Q: You have the region bounded by y = x² and y = x, and you need to revolve it about the x-axis. Which method do you use if you want to integrate with respect to x?
A: Disk/washer method, because the axis of revolution (x-axis) matches the variable of integration (x). The outer radius is x (the higher curve on [0, 1]) and the inner radius is x².
Q: Same region, but now revolve about the y-axis and integrate with respect to x. Which method?
A: Shell method, because the axis of revolution (y-axis) differs from the variable of integration (x). The shell radius is x and the shell height is x − x².
Q: When finding the area between two curves, why might you choose to integrate with respect to y instead of x?
A: When the curves are more naturally expressed as functions of y, or when integrating with respect to x would require splitting into multiple separate integrals due to the curves crossing or the region's shape.
Q: The region between y = √x and y = 0 from x = 0 to x = 4 is revolved about the line y = −1. How does this change the washer setup compared to revolving about the x-axis?
A: The outer radius becomes √x − (−1) = √x + 1, and the inner radius becomes 0 − (−1) = 1. Every distance is now measured from the line y = −1, not from y = 0.
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