--- title: "AP Calculus 8.9: Disc Method Around the x- or y-Axis" description: "Review AP Calc 8.9 disc method for volumes of revolution, including x-axis and y-axis setup, definite integrals, radius, bounds, dx, and dy." canonical: "https://fiveable.me/ap-calc/unit-8/volume-with-disc-method-revolving-around-x-or-y-axis/study-guide/ZYnKJoUmSzzGYJHCSqM3" type: "study-guide" subject: "AP Calculus AB/BC" unit: "Unit 8 – Applications of Integration" lastUpdated: "2026-06-09" --- # AP Calculus 8.9: Disc Method Around the x- or y-Axis ## Summary Review AP Calc 8.9 disc method for volumes of revolution, including x-axis and y-axis setup, definite integrals, radius, bounds, dx, and dy. ## Guide The [disc method](/ap-calc/unit-8/volume-with-disc-method-revolving-around-other-axes/study-guide/q8HRD4jQBvvU4ZRRkoTb "fv-autolink") finds the volume of a solid made by revolving a region around the x- or y-axis. You slice the solid into thin discs perpendicular to the axis, write each disc's volume as $\pi(\text{radius})^2$ times its thickness, then integrate. For AP Calculus, match the slice direction to the axis of rotation before choosing $dx$ or $dy$. ## Why This Matters for the AP Calculus Exam Volumes of revolution show up in [Unit 8](/ap-calc/unit-8 "fv-autolink"), Applications of Integration, which carries a noticeable share of the AP Calculus exam (more weight on AB than BC). This topic asks you to calculate [volumes of solids of revolution](/ap-calc/key-terms/solids-of-revolution "fv-autolink") using definite integrals. On the exam you will need to recognize a region, decide whether to integrate with respect to $x$ or $y$, set up a correct integral with proper notation, and evaluate it. Free-response questions in this unit reward a clearly written integral expression even before you compute, so practice writing the full setup with $\pi$, the squared radius, the bounds, and the correct differential. Getting comfortable here also sets you up for the disc method around other axes (8.10) and the [washer method](/ap-calc/unit-8/volume-with-washer-method-revolving-around-other-axes/study-guide/LlG9jCFLxe4kpDTDNtHX "fv-autolink") (8.11 and 8.12). ## Key Takeaways - Revolving around the x-axis: $V = \pi\int_a^b (f(x))^2\,dx$, where the radius is $f(x)$ and the thickness is $dx$. - Revolving around the y-axis: $V = \pi\int_c^d (f(y))^2\,dy$, where the radius is $f(y)$ and the thickness is $dy$. - The discs are sliced perpendicular to the axis of rotation, so the axis decides whether you integrate in $x$ or $y$. - These formulas work only when one curve forms the radius and the solid has no hole. A hole means you need the washer method. - For y-axis rotation, rewrite the curve as $x = f(y)$ before setting up the integral. - Always sketch the region and shade it before writing the integral so you pick the right bounds. ## Volumes of Solids of Revolution When you find the volume of a [solid of revolution](/ap-calc/key-terms/solid-of-revolution "fv-autolink"), you are measuring how much space a 3D shape occupies. You take a curve, rotate it around an axis, and that rotation sweeps out a solid. The disc method calculates the volume of that solid by adding up many thin slices. ### The Disc Method: X-Axis The disc method slices the solid into infinitely thin discs perpendicular to the axis of rotation. Adding the volumes of these discs with a definite integral gives the total volume. To find the volume of a solid rotated around the x-axis, you sum the volumes of many thin cross-sections. Each cross-section has a width of $dx$ (approaching 0) and a radius of $f(x)$. Each one is a very flat cylinder, with volume $w \times \pi r^2$, where $w$ is the width and $r$ is the radius. Plugging in the width and radius gives: $$ \pi (f(x))^2 dx $$ To add all of these volumes together, you use an integral: $$ \int_{a}^{b}\pi (f(x))^2dx $$ Here $a$ and $b$ are the boundaries for $f(x)$, given as $x = a$ and $x = b$. ### The Disc Method: Y-Axis Sometimes you rotate a region around the y-axis instead. The process is the same, except you replace $f(x)$ and $dx$ with $f(y)$ and $dy$: $$ \int_{c}^{d}\pi (f(y))^2dy $$ The boundaries $c$ and $d$ are given as $y = c$ and $y = d$. ### Solving Using the Disc Method To solve a rotated-solid problem around the x- or y-axis with just one equation, use these steps: 1. **Determine the Axis**: Identify whether the solid revolves around the x-axis or y-axis. This decides how you set up the integral. 2. **Slice the Solid**: Picture slicing the solid into thin discs perpendicular to that axis. Each disc is a tiny volume element. 3. **Set Up the Integral**: Use a definite integral to sum the volumes of all the discs along the [interval](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink"), using the formula above. 4. **Evaluate the Integral**: Once it is set up, evaluate to find the total volume. > Important: The formula above works only for solids rotated around the x- or y-axis when a single $f(x)$ or $f(y)$ equation is given. For rotating around other axes or using two equations with the washer method, check out the [8.10](/ap-calc/unit-8/volume-disc-method-revolving-around-other-axes/study-guide/q8HRD4jQBvvU4ZRRkoTb), [8.11](/ap-calc/unit-8/volume-washer-method-revolving-around-x-or-y-axis/study-guide/9kgWFLHEU5oAfAA5aXaq), and [8.12](/ap-calc/unit-8/volume-washer-method-revolving-around-other-axes/study-guide/LlG9jCFLxe4kpDTDNtHX) guides. --- ## Practicing with the Disc Method Here is how to apply these steps to practice problems. ### Disc Method: Practice Question 1 Calculate the volume of the solid obtained by revolving the region bounded by the curves $y=x^2$, $x=1$, and the y-axis about the x-axis. **Step 1: Determine the Axis** The question revolves the region about the x-axis, so you will integrate with respect to $x$. **Step 2: Slice the Solid** Picture slicing the region into thin discs perpendicular to the x-axis. Graph all of your equations and shade the [area](/ap-calc/unit-6/applying-properties-definite-integrals/study-guide/lUbcVbDG5QVysAn9 "fv-autolink") to be revolved. **Step 3: Set Up the Integral** Since you are revolving around the x-axis, integrate with respect to $x$ and use the general formula, where $f(x)$ defines the region and $a$ and $b$ are the interval of integration. $$ V= \int_{a}^{b}\pi (f(x))^2dx=\pi\int_{a}^{b}(f(x)^2)\, dx $$ **Step 4: Evaluate the Integral** Integrate $y=x^2$ from $x=0$ to $x=1$. The lower bound is where $y=x^2$ meets the y-axis at $x=0$, and the upper bound is at $x=1$. $$ V=\pi\int_{0}^{1}(x^2)^2\, dx $$ $$ V=\pi\int_{0}^{1}x^4\, dx $$ $$ V=\pi[\frac15x^5]\vert_0^1 $$ $$ V=\pi(\frac15(1)^5-\frac15(0)^5) $$ $$ V=\pi(\frac15)=\boxed{\frac{\pi}{5}} $$ Now try another one. ### Disc Method: Practice Question 2 Find the volume of the solid found by rotating the region bounded by $y = 1$, $y=8$, and $y = x^3$ around the y-axis. **Step 1: Determine the Axis** You are rotating around the y-axis, so your integral should be in terms of $f(y)$ instead of $f(x)$. **Step 2: Slice the Solid** Draw out the region. You are rotating the area between the curve and the y-axis, bounded above and below by the horizontal lines $y = 8$ and $y = 1$. **Step 3: Set Up the Integral** Be careful here. For rotating around the y-axis, your [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") needs to be in terms of $f(y)$, meaning $x = f(y)$. The given equation is: $$ y = x^3 $$ To rewrite this in terms of $y$, take the cube root of both sides: $$ \sqrt[3] {y}=x $$ This is the correct function to use in the general integral formula. **Step 4: Evaluate the Integral** The general format is: $$ V= \int_{a}^{b}\pi (f(y))^2dy=\pi\int_{a}^{b}(f(y)^2)\, dy $$ The bounds are $y = 1$ and $y=8$. Plugging in: $$ V= \int_{1}^{8}\pi (\sqrt[3]{y})^2dy=\pi\int_{1}^{8}(\sqrt[3]{y}^2)\, dy=\pi\int_{1}^{8}({y}^{2/3}) dy $$ This simplifies to: $$ V=\pi[\frac35y^{5/3}]\vert_1^8 $$ $$ V=\pi(\frac35(8)^{5/3}-\frac35(1)^{5/3}) $$ $$ V=\pi(\frac35(32)-\frac35(1)) $$ $$ V=\pi(\frac35(31)) $$ $$ V=\boxed{\frac{93}5 \pi} $$ --- ## How to Use This on the AP Calculus Exam ### Free Response Write the full integral setup before you simplify. A clear expression like $V=\pi\int_{a}^{b}(f(x))^2\,dx$ with correct bounds and differential shows your reasoning and is important for clear exam work. If the region is revolved around the y-axis, show that you rewrote the curve as $x = f(y)$. ### Problem Solving - Sketch and shade the region first. The picture tells you the radius and the bounds. - Match the differential to the axis: $dx$ for x-axis rotation, $dy$ for y-axis rotation. - Square the entire radius, not just part of it. For $\sqrt[3]{y}$, squaring gives $y^{2/3}$. - After integrating, plug in both bounds and subtract carefully. ### Common Trap If the region is revolved around the y-axis but your function is written as $y = f(x)$, you cannot just swap letters. Solve for $x$ first so your radius is a function of $y$. ## Common Misconceptions - **Forgetting to square the radius.** The disc area is $\pi r^2$, so the [integrand](/ap-calc/key-terms/integrand "fv-autolink") must include $(f(x))^2$ or $(f(y))^2$, not just $f(x)$ or $f(y)$. - **Using the wrong differential.** Rotating around the x-axis uses $dx$ and bounds in $x$. Rotating around the y-axis uses $dy$ and bounds in $y$. Mixing them gives the wrong setup. - **Using disc when there is a hole.** The disc method assumes the solid is solid all the way through. If the region does not touch the axis of rotation, there is a gap and you need the washer method. - **Confusing volume with area.** This topic finds volume, not [area between curves](/ap-calc/key-terms/area-between-curves "fv-autolink"). The $\pi$ and the squared radius are what turn the slice into a 3D disc. - **Not converting for y-axis rotation.** When you rotate around the y-axis, rewrite the curve as $x = f(y)$ before plugging into the formula. ## Related AP Calculus Guides - [Unit 8 Overview: Applications of Integration](/ap-calc/unit-8/review/study-guide/95uuVjdtA80roOMvV8IK) - [8.1 Finding the Average Value of a Function on an Interval](/ap-calc/unit-8/finding-average-value-function-on-an-interval/study-guide/HjiYTRAnQdY0eCQpqtpg) - [8.7 Volumes with Cross Sections: Squares and Rectangles](/ap-calc/unit-8/volumes-with-cross-sections-squares-rectangles/study-guide/djttfP0mZkJ7Nn8QrB7r) - [8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals](/ap-calc/unit-8/connecting-position-velocity-acceleration-functions-using-integrals/study-guide/k9tY28YXs7YDVu1uqFuw) - [8.4 Finding the Area Between Curves Expressed as Functions of x](/ap-calc/unit-8/finding-area-between-curves-expressed-as-functions-x/study-guide/Zyj7XJuPfoWBuAJ96ZAG) - [8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts](/ap-calc/unit-8/using-accumulation-functions-definite-integrals-applied-contexts/study-guide/nUlJKvXqRcsfLnVMd5fG) ## Vocabulary - **definite integral**: The integral of a function over a specific interval [a, b], representing the net signed area between the curve and the x-axis. - **disc method**: A technique for finding the volume of a solid of revolution by integrating the cross-sectional areas of circular discs perpendicular to the axis of rotation. - **solids of revolution**: Three-dimensional solids formed by rotating a two-dimensional region around an axis. ## FAQs ### What is the disc method in AP Calculus? The disc method finds the volume of a solid formed by revolving a region around an axis. You treat each cross section as a thin disc with area pi(radius)^2 and add the discs with a definite integral. ### What is the disc method formula around the x-axis? For a region revolved around the x-axis with radius f(x), use V = pi int_a^b [f(x)]^2 dx. The bounds are x-values, and the slices are perpendicular to the x-axis. ### What is the disc method formula around the y-axis? For a region revolved around the y-axis with radius f(y), use V = pi int_c^d [f(y)]^2 dy. Rewrite the curve in terms of y when needed, and use y-values for the bounds. ### How do you know whether to use dx or dy? Slice perpendicular to the axis of rotation. Around the x-axis, you usually integrate with dx; around the y-axis, you usually integrate with dy. A sketch helps you confirm the radius and bounds. ### When do you need washer method instead of disc method? Use washer method when the rotated region creates a hole, meaning there is an outer radius and an inner radius. Disc method works when the cross section is filled in with one radius. ### How is AP Calc 8.9 tested? AP Calc 8.9 typically tests whether you can set up a correct definite integral for a solid of revolution around the x-axis or y-axis, including pi, the squared radius, bounds, and the right differential. ## Structured Data ```json {"@context":"https://schema.org","@type":"FAQPage","inLanguage":"en","mainEntity":[{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-8/volume-with-disc-method-revolving-around-x-or-y-axis/study-guide/ZYnKJoUmSzzGYJHCSqM3#what-is-the-disc-method-in-ap-calculus","name":"What is the disc method in AP Calculus?","acceptedAnswer":{"@type":"Answer","text":"The disc method finds the volume of a solid formed by revolving a region around an axis. 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