From a Base Slice to a Quarter-Circle Area
A solid can be described by the shapes of its cross sections: slices made by planes perpendicular to a chosen axis. To find its volume, first determine the area of a typical cross section from the corresponding segment in the base region. For quarter-circle cross sections, the crucial question is what that segment represents.
In this tutorial, the segment in the base region gives the radius of each quarter circle. It is not the diameter. Once its length is written as a function of the slice position, the cross-sectional area is one quarter of the area of a full disk with that radius.
The area of a full disk of radius \(r\) is \(\pi r^2\). A quarter circle is one of four congruent sectors that together make that disk, so its area is one fourth of the full-disk area. This is why the factor is \(\frac{\pi}{4}\), not \(\frac{\pi}{2}\) or \(\pi\).
By the slicing principle, the volume is the integral of cross-sectional area over the interval of slice positions. Thus, if slices perpendicular to the \(x\)-axis have radius \(r(x)\) for \(a\leq x\leq b\), then
For slices perpendicular to the \(y\)-axis, the corresponding expression is \(\frac{\pi}{4}\int_c^d [r(y)]^2\,dy\). The integration variable and bounds must describe the direction and extent of the slices. The radius must be a nonnegative length on the interval of integration.
Finding the Radius from the Base Region
For vertical slices, the base-region segment runs from a lower boundary \(y=g(x)\) to an upper boundary \(y=f(x)\). Its length is \(f(x)-g(x)\), provided the upper boundary is at least as high as the lower boundary throughout the interval. If this segment is specified as the radius, then \(r(x)=f(x)-g(x)\).
For horizontal slices, the segment runs from a left boundary \(x=L(y)\) to a right boundary \(x=R(y)\). Its length is \(R(y)-L(y)\), provided the right boundary is at least as far right as the left boundary. If it gives the radius, then \(r(y)=R(y)-L(y)\).
In either direction, the volume integrand squares the segment length. That does not mean the segment may be treated as a diameter: the area formula depends on whether the segment represents a radius or a diameter before it is squared.
Use \(dx\) for slices perpendicular to the \(x\)-axis and \(dy\) for slices perpendicular to the \(y\)-axis.
For vertical slices, subtract lower \(y\) from upper \(y\). For horizontal slices, subtract left \(x\) from right \(x\).
Because the segment gives the radius, square its length and multiply by \(\frac{\pi}{4}\) to obtain the cross-sectional area.
Choose bounds that cover the base region once in the selected direction.
Worked Examples
Worked Example: Vertical Slices Through a Quarter-Disk Base
The base region is bounded by \(y=0\), \(x=0\), and \(x^2+y^2=4\) in the first quadrant. Cross sections perpendicular to the \(x\)-axis are quarter circles, and the vertical segment in the base gives each radius. Find the volume.
First, solve the circle equation for the upper boundary. Since the region is in the first quadrant, \(y\geq 0\), so \(y=\sqrt{4-x^2}\). The vertical slice extends from \(y=0\) to \(y=\sqrt{4-x^2}\), and therefore
The \(x\)-values run from \(0\) to \(2\). Square the radius in the quarter-circle area formula, then integrate:
Check the antiderivative: \(\frac{d}{dx}(4x-\frac{x^3}{3})=4-x^2\), which is the integrand. The volume is \(\frac{4\pi}{3}\) cubic units.
Worked Example: A Triangular Base with Radius \(x\)
The base region is bounded by \(y=0\), \(y=x\), and \(x=2\). Cross sections perpendicular to the \(x\)-axis are quarter circles, and the vertical segment in the base gives each radius. Find the volume.
At a fixed \(x\) between \(0\) and \(2\), the segment extends from \(y=0\) to \(y=x\). Thus \(r(x)=x-0=x\), and the bounds are \(0\leq x\leq 2\). The volume is
Check the antiderivative: \(\frac{d}{dx}(\frac{x^3}{3})=x^2\), the integrand. The volume is \(\frac{2\pi}{3}\) cubic units. The radius varies from \(0\) at \(x=0\) to \(2\) at \(x=2\), as the base-region slice does.
Worked Example: Horizontal Slices and a Radius Depending on \(y\)
The base region is bounded by \(x=0\), \(y=x\), and \(y=3\). Cross sections perpendicular to the \(y\)-axis are quarter circles, and the horizontal segment in the base gives each radius. Find the volume.
For a fixed \(y\) between \(0\) and \(3\), the segment runs from \(x=0\) to \(x=y\), because the line \(y=x\) can be written \(x=y\). Therefore the radius is \(r(y)=y-0=y\). Integrating with respect to \(y\) gives
Check the antiderivative: \(\frac{d}{dy}(\frac{y^3}{3})=y^2\), the integrand. The volume is \(\frac{9\pi}{4}\) cubic units. Here the bounds and radius are expressed in terms of \(y\), since the slices are horizontal.
Why the Factor Matters
The full-disk area, semicircle area, and quarter-circle area use different factors. When the stated base segment is a radius, the quarter-circle formula is the relevant one:
| Cross section | Meaning of segment \(r\) | Area |
|---|---|---|
| Full circle | Radius | \(\pi r^2\) |
| Semicircle | Radius | \(\frac{\pi r^2}{2}\) |
| Quarter circle | Radius | \(\frac{\pi r^2}{4}\) |
The diameter convention is different. If a segment of length \(d\) is the diameter of a quarter circle, the radius is \(d/2\), so its area is \(\frac{\pi}{4}(d/2)^2=\frac{\pi d^2}{16}\). Do not substitute a diameter directly as though it were the radius.
Setting Up Accurately
A reliable setup makes each geometric choice explicit. First state which way the slices are taken. Then identify the endpoints of a typical segment in the base and subtract them in the order that gives a nonnegative length. Use that length as \(r\), apply \(\frac{\pi r^2}{4}\), and integrate across the full interval of slices.
The square in the formula is essential: if the radius is a function such as \(r(x)=\sqrt{4-x^2}\), its square simplifies to \(4-x^2\). Keep the square until after the radius has been correctly identified. It is not a substitute for checking the segment’s geometric meaning.
The area of each slice has units of square units. Integrating that area with respect to a length variable gives cubic units, as a volume should. This is a useful dimensional check, but it does not replace checking the factor \(\frac{\pi}{4}\), the radius, and the bounds.
Check Your Understanding
For each question, identify the radius and the correct quarter-circle area before proceeding to a volume integral.
- A vertical base segment extends from \(y=1\) to \(y=3x\), where \(3x\geq 1\). If it gives the radius, what is \(r(x)\), and what is the cross-sectional area?
- A horizontal segment has length \(5-y\) and gives the radius. Write the quarter-circle area as a function of \(y\).
- What factor should multiply \(r^2\) for a quarter-circle cross section whose segment gives the radius?
- If a base segment gives the diameter \(d\), what is the radius and the quarter-circle area in terms of \(d\)?
- For horizontal slices with radius \(r(y)\) on \(c\leq y\leq d\), write the volume integral.
