Answer: $3$

Find vertices of the region: $y = x$ and $y = 6 - 2x$ intersect when $x = 6-2x \Rightarrow x = 2$, so $(2,2)$. $y = x$ meets $y = 0$ at $(0,0)$. $y = 6-2x$ meets $y = 0$ at $(3,0)$. Set up the integral: From $x = 0$ to $x = 2$, the top is $y = x$ and bottom is $y = 0$. From $x = 2$ to $x = 3$, the top is $y = 6-2x$ and bottom is $y = 0$. Area $= \int_0^2 x\,dx + \int_2^3(6-2x)\,dx$. Evaluate: $\int_0^2 x\,dx = 2$. $\int_2^3(6-2x)\,dx = [6x - x^2]_2^3 = (18-9) - (12-4) = 9 - 8 = 1$. Total $= 2 + 1 = 3$. Why distractors fail: Option A ($2$) only computes the first piece. Option C ($4$) may arise from a limit error. Option D ($6$) likely treats the region as a $3 \times 2$ rectangle.

Answer: The integral gives the net signed area; since $y = x^2 - 2x < 0$ on part of $[1, 3]$, the actual area requires splitting the integral at $x = 2$ and using absolute values, giving area $= 2$

Check where $x^2 - 2x = 0$: $x(x-2) = 0$, so $x = 0$ and $x = 2$. The function is negative on $(0, 2)$ and positive on $(2, \infty)$. Compute the actual area: Area $= \int_1^2 |x^2-2x|\,dx + \int_2^3 (x^2-2x)\,dx = \int_1^2 (2x-x^2)\,dx + \int_2^3(x^2-2x)\,dx$. First piece: $\left[x^2 - \frac{x^3}{3}\right]_1^2 = (4-\frac{8}{3}) - (1-\frac{1}{3}) = \frac{4}{3} - \frac{2}{3} = \frac{2}{3}$. Second piece: $\left[\frac{x^3}{3} - x^2\right]_2^3 = (9-9)-(\frac{8}{3}-4) = 0 + \frac{4}{3} = \frac{4}{3}$. Total area $= \frac{2}{3} + \frac{4}{3} = 2$. Why distractors fail: Option A ignores the sign issue. Option C is wrong—$x = 2$ is a root in $[1,3]$. Option D's calculation is incorrect; $\int_1^3(x^2-2x)\,dx = \frac{2}{3}$ is correct as a signed integral.

Answer: Both methods give the same volume when set up correctly

Equivalence of methods: The disk/washer method and the shell method are two different approaches to computing the same volume. When set up correctly, they yield identical results. Washer approach (Student A): $V = \pi\int_{-2}^{2}[4^2 - (x^2)^2]\,dx = \pi\int_{-2}^{2}(16 - x^4)\,dx$. Shell approach (Student B): $V = 2\pi\int_0^4 y \cdot 2\sqrt{y}\,dy = 4\pi\int_0^4 y^{3/2}\,dy$. Both evaluate to $\frac{256\pi}{5}$. Why distractors fail: Options A and C incorrectly claim only one method works. Option D incorrectly states different variables produce different results.

Answer: $\int_0^2 (2x - x^2)\,dx$ and $\int_0^4 \left(\sqrt{y} - \frac{y}{2}\right)\,dy$

Find intersection points: $x^2 = 2x \Rightarrow x(x-2) = 0$, so $x = 0, 2$. In $y$: when $x = 2$, $y = 4$; when $x = 0$, $y = 0$. Set up the $x$-integral: On $[0,2]$, $2x \geq x^2$, so area $= \int_0^2(2x - x^2)\,dx$. Set up the $y$-integral: Solving for $x$: $y = x^2 \Rightarrow x = \sqrt{y}$ (right) and $y = 2x \Rightarrow x = y/2$ (left). On $[0,4]$, $\sqrt{y} \geq y/2$, so area $= \int_0^4(\sqrt{y} - y/2)\,dy$. Why distractors fail: Option B reverses the subtraction order, giving negative values. Option C uses incorrect limits ($0$ to $4$ for $x$ and $0$ to $2$ for $y$). Option D uses the wrong upper $y$-limit.

Answer: When the axis of rotation is perpendicular to the slicing direction used in the disk/washer approach, making it difficult to express the radius as a function of the integration variable

Choosing between methods: The shell method is advantageous when using the washer method would require solving for $x$ in terms of $y$ (or vice versa) or when the resulting expressions would need to be split into multiple integrals. Why the correct answer works: Option A correctly identifies the key criterion: when the natural variable of integration aligns with the axis of rotation, shells produce parallel slices that are easier to express. Why distractors fail: Option B is irrelevant; both methods can handle multiple boundary curves. Option C is too restrictive—shells can be used about any axis. Option D has nothing to do with the choice of method.

Answer: $\dfrac{32}{3}$

Find intersection points: Set $x^2 = 4$, so $x = -2$ and $x = 2$. Set up the integral: Since $4 \geq x^2$ on $[-2, 2]$, the area is $\int_{-2}^{2} (4 - x^2)\,dx$. Evaluate: $\int_{-2}^{2}(4 - x^2)\,dx = \left[4x - \frac{x^3}{3}\right]_{-2}^{2} = \left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right) = \frac{16}{3} + \frac{16}{3} = \frac{32}{3}$. Why distractors fail: Option A ($\frac{16}{3}$) is only half the area (integrating from $0$ to $2$). Option C ($\frac{8}{3}$) results from an integration error. Option D ($16$) incorrectly treats the area as a rectangle $4 \times 4$.

Answer: $V = 2\pi \int_a^b x\,f(x)\,dx$

Shell method formula: The shell method uses cylindrical shells with circumference $2\pi r$, height $h$, and thickness $dx$. When revolving about the $y$-axis, the radius is $x$ and the height is $f(x)$. Why the correct answer works: Option B gives $V = 2\pi \int_a^b x\,f(x)\,dx$, correctly representing the product of shell circumference ($2\pi x$), height ($f(x)$), and thickness ($dx$). Why distractors fail: Option A is the disk method formula for rotation about the $x$-axis. Option C incorrectly squares $f(x)$ and includes the $2\pi$ factor. Option D is missing the factor of 2 in $2\pi$.

Answer: $\pi\int_0^{2\pi} (2 + \sin(y))^2\,dy$

Volume by known cross-sections: When the cross-section at height $y$ is a disk of radius $r(y)$, the area is $\pi[r(y)]^2$. Summing these slices gives $V = \pi\int_0^{2\pi}(2 + \sin(y))^2\,dy$. Why the correct answer works: Option B correctly applies the formula for volume using circular cross-sections: $\int A(y)\,dy$ where $A(y) = \pi r(y)^2$. Why distractors fail: Option A uses the shell method formula, which does not apply here. Option C expands incorrectly—$(2 + \sin y)^2 = 4 + 4\sin y + \sin^2 y$, not $4 + \sin^2 y$. Option D computes circumference times height, not area times height.

Answer: $8\pi$

Identify the radius: Revolving about the $x$-axis, the radius of each disk is $R(x) = \sqrt{x}$. Set up and evaluate the integral: $V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi \cdot \frac{16}{2} = 8\pi$. Why distractors fail: Option A ($4\pi$) results from forgetting to square the radius or from an upper limit error. Option C ($16\pi$) results from not dividing by 2 in the antiderivative. Option D ($2\pi$) likely confuses factors from the shell method.

Answer: $\int_a^b [f(x) - g(x)]\,dx$

Area between curves formula: The area between two curves is found by integrating the difference of the top function minus the bottom function: $\int_a^b [f(x) - g(x)]\,dx$ when $f(x) \geq g(x)$. Why the correct answer works: Option B correctly subtracts the lower curve from the upper curve, yielding a non-negative integrand whose integral is the enclosed area. Why distractors fail: Option A adds the functions, which does not represent the vertical distance between them. Option C reverses the subtraction and would give a negative result. Option D multiplies two separate integrals, which has no geometric meaning for area between curves.

Answer: No; when integrating with respect to $y$, we integrate the right function minus the left function

Horizontal slicing vs. vertical slicing: When slicing horizontally (integrating with respect to $y$), each strip extends from the leftmost curve to the rightmost curve. The width of each strip is (right $-$ left), not (top $-$ bottom). Why the correct answer works: Option B correctly identifies that horizontal slicing requires $\int [x_{\text{right}}(y) - x_{\text{left}}(y)]\,dy$. Why distractors fail: Options A and D incorrectly claim top minus bottom works for both variables. Option C incorrectly introduces squaring, which applies to volume of revolution, not area.

Answer: Evaluate $\int_a^b |f(x)|\,dx$ or equivalently $-\int_a^b f(x)\,dx$

Handling negative functions: When $f(x) < 0$, the integral $\int_a^b f(x)\,dx$ yields a negative value. Area is always non-negative, so we take the absolute value of the integrand. Why the correct answer works: Option B uses $|f(x)|$, which equals $-f(x)$ when $f(x) < 0$, ensuring the integrand is positive and the result is the actual geometric area. Why distractors fail: Option A is wrong because the integral of a negative function is negative, not the area. Option C squares the function, which changes the geometric meaning entirely. Option D reverses both the sign of the integral and the direction, producing the same negative value.

Answer: Because integration is a linear operation and does not distribute over squaring

Linearity of integration: Integration is linear: $\int(cf + g) = c\int f + \int g$. However, $\int f^2 \neq (\int f)^2$ because squaring is not a linear operation. Why the correct answer works: Option A correctly identifies the fundamental algebraic reason: the integral operator does not commute with the squaring operation. Why distractors fail: Option B is false—both use the same limits. Option C is not always true (consider $f(x) = 0$). Option D gives a specific geometric interpretation that doesn't explain the general algebraic inequality.

Answer: The area of the region between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$

Geometric meaning of the definite integral: When $f(x) \geq 0$ on $[a,b]$, the definite integral $\int_a^b f(x)\,dx$ computes the signed area between the curve $y = f(x)$ and the $x$-axis. Why the correct answer works: Option B directly states the geometric interpretation: the area under the non-negative curve and above the $x$-axis. Why distractors fail: Option A describes a derivative-related concept (slope), not an integral. Option C describes arc length, which requires a different integral formula. Option D describes $\frac{1}{b-a}\int_a^b f(x)\,dx$, not the integral itself.

Answer: $\dfrac{8\pi}{3}$

Find intersection points: $\sqrt{x} = x/2 \Rightarrow 2\sqrt{x} = x \Rightarrow 4x = x^2 \Rightarrow x(x-4) = 0$, so $x = 0$ and $x = 4$. Set up the washer integral: On $[0,4]$, $\sqrt{x} \geq x/2$. Outer radius $R = \sqrt{x}$, inner radius $r = x/2$. $V = \pi\int_0^4 [x - x^2/4]\,dx$. Evaluate: $V = \pi\left[\frac{x^2}{2} - \frac{x^3}{12}\right]_0^4 = \pi\left(8 - \frac{64}{12}\right) = \pi\left(8 - \frac{16}{3}\right) = \pi \cdot \frac{8}{3} = \frac{8\pi}{3}$. Why distractors fail: Option B doubles the answer. Option C halves it. Option D arises from misidentifying the bounds or using $(\sqrt{x} - x/2)^2$ instead of $x - x^2/4$.

Answer: $\dfrac{5}{2}$

Split at the point where the expression changes sign: $|x - 1| = -(x-1)$ when $x < 1$ and $(x-1)$ when $x \geq 1$. So we split: $\int_0^1 (1-x)\,dx + \int_1^3 (x-1)\,dx$. Evaluate each part: $\int_0^1(1-x)\,dx = \left[x - \frac{x^2}{2}\right]_0^1 = \frac{1}{2}$. $\int_1^3(x-1)\,dx = \left[\frac{x^2}{2} - x\right]_1^3 = (\frac{9}{2}-3) - (\frac{1}{2}-1) = \frac{3}{2} + \frac{1}{2} = 2$. Total $= \frac{1}{2} + 2 = \frac{5}{2}$. Why distractors fail: Option A ($\frac{3}{2}$) may come from integrating without absolute value. Option C ($2$) is only the second piece. Option D ($\frac{7}{2}$) adds incorrectly.

Answer: $\pi\int_0^1 [(2 - x^2)^2 - (2 - x)^2]\,dx$

Identify radii from the axis $y = 2$: The axis is above the region. The outer radius (farther from $y = 2$) is $R = 2 - x^2$ (distance from $y = 2$ to the lower curve $y = x^2$). The inner radius is $r = 2 - x$ (distance from $y = 2$ to the upper curve $y = x$). Set up the integral: $V = \pi\int_0^1[(2-x^2)^2 - (2-x)^2]\,dx$. Why distractors fail: Option B swaps outer and inner radii. Option C uses $(x-2)$ and $(x^2-2)$, which are negative—while squaring fixes the sign, the outer/inner assignment is wrong. Option D is a shell method setup, not washers.

Answer: With respect to $y$; area $= \dfrac{32}{3}$

Choose the variable of integration: The curve $x = y^2$ is naturally expressed as a function of $y$. Horizontal slicing (integrating with respect to $y$) avoids splitting the integral. Find limits and set up: Intersection: $y^2 = 4 \Rightarrow y = -2, 2$. The right function is $x = 4$ and left is $x = y^2$. Area $= \int_{-2}^{2}(4 - y^2)\,dy$. Evaluate: $\int_{-2}^{2}(4 - y^2)\,dy = \left[4y - \frac{y^3}{3}\right]_{-2}^{2} = \left(8 - \frac{8}{3}\right) - \left(-8 + \frac{8}{3}\right) = \frac{32}{3}$. Why distractors fail: Option A gives the right idea of $x$ integration but the wrong value—using $x$ would require $\int_0^4 2\sqrt{x}\,dx$, which also gives $\frac{32}{3}$, but the option states $\frac{16}{3}$. Option C has the correct variable but the wrong value. Option D has the wrong variable listed.

Answer: $8\pi$

Set up the shell method integral: Revolving about the $y$-axis with vertical shells: radius $= x$, height $= x^2$, thickness $= dx$. So $V = 2\pi\int_0^2 x \cdot x^2\,dx = 2\pi\int_0^2 x^3\,dx$. Evaluate the integral: $V = 2\pi\left[\frac{x^4}{4}\right]_0^2 = 2\pi \cdot \frac{16}{4} = 2\pi \cdot 4 = 8\pi$. Why distractors fail: Option A ($\frac{32\pi}{5}$) comes from using $x^4$ instead of $x^3$ in the integrand. Option C ($\frac{16\pi}{3}$) results from integrating $x^2$ instead of $x^3$. Option D ($4\pi$) omits the $2\pi$ factor or halves the result.

Answer: $\dfrac{1}{2}$

Analyze the sign of $x^3 - x$: $x^3 - x = x(x-1)(x+1)$. This is $\geq 0$ on $[-1, 0]$ and $\leq 0$ on $[0, 1]$. Use absolute value and split: Area $= \int_{-1}^{0}(x^3 - x)\,dx + \int_0^1 -(x^3 - x)\,dx = \int_{-1}^{0}(x^3 - x)\,dx + \int_0^1(x - x^3)\,dx$. Evaluate: $\int_{-1}^{0}(x^3 - x)\,dx = \left[\frac{x^4}{4} - \frac{x^2}{2}\right]_{-1}^0 = 0 - (\frac{1}{4} - \frac{1}{2}) = \frac{1}{4}$. $\int_0^1(x - x^3)\,dx = \left[\frac{x^2}{2} - \frac{x^4}{4}\right]_0^1 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$. Total $= \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$. Why distractors fail: Option A ($0$) is the signed integral $\int_{-1}^1(x^3-x)\,dx$, which cancels due to symmetry. Option C ($\frac{1}{4}$) counts only one of the two pieces. Option D ($1$) is an overcount.

Answer: Because the parabola opens leftward, making horizontal slicing require only a single integral, while vertical slicing would require splitting at the vertex

Analyze the geometry: The curve $x = 4 - y^2$ is a leftward-opening parabola. With vertical slices, the top/bottom boundary changes at the vertex, requiring two separate integrals. With horizontal slices, one integral suffices: $\int (\text{right} - \text{left})\,dy$. Why the correct answer works: Option A correctly identifies the geometric reason: horizontal slicing avoids splitting the integral at the point where the boundary changes. Why distractors fail: Option B is false—both approaches give exact results. Option C is incorrect; $x = 4 - y^2$ can be solved for $y$ as $y = \pm\sqrt{4-x}$. Option D confuses the shell method with the choice of integration variable for area.

Answer: $2\pi\int_1^{e} x\ln(x)\,dx$

Choose the method: Revolving about the $y$-axis with the region described in terms of $x$: the shell method integrates with respect to $x$. Set up shell components: Shell radius $= x$, shell height $= \ln(x)$, limits from $x = 1$ to $x = e$. So $V = 2\pi\int_1^e x\ln(x)\,dx$. Why distractors fail: Option A is the disk method about the $x$-axis with incorrect limits. Option C is the disk method about the $x$-axis (not the $y$-axis). Option D uses a valid but different approach—shells about the $x$-axis—and has incorrect components for this problem.

Answer: The washer method accounts for a hole in each cross-section by subtracting the inner radius squared from the outer radius squared

Disk vs. washer: The disk method produces solid circular cross-sections. When there is a gap between the region and the axis of rotation, each cross-section is a washer (annulus) with an outer radius $R$ and inner radius $r$. Why the correct answer works: Option B correctly describes the washer method: $V = \pi\int_a^b([R(x)]^2 - [r(x)]^2)\,dx$, subtracting the inner disk from the outer disk. Why distractors fail: Option A is incorrect because both methods can use $dx$ or $dy$. Option C describes the shell method, not the washer method. Option D is false since the washer method works for rotation about any axis.

Answer: The radius of the disk at position $x$, measured from the axis of rotation to the curve

Understanding the disk method: The disk method models each cross-section as a circular disk. The area of a circle is $\pi r^2$, where $r$ is the radius. Why the correct answer works: Option B correctly identifies $R(x)$ as the distance from the axis of rotation to the boundary curve—the radius of the circular cross-section. Why distractors fail: Option A describes circumference ($2\pi r$), not the quantity squared in the formula. Option C describes the diameter ($2r$), which would need to be halved before squaring. Option D describes $dx$, the infinitesimal thickness, not $R(x)$.

Answer: $\pi\int_0^1 (x^2 - x^6)\,dx$

Find intersection points: $x = x^3 \Rightarrow x^3 - x = 0 \Rightarrow x(x^2-1) = 0$, so $x = 0$ and $x = 1$ in the first quadrant. Identify outer and inner radii: On $[0,1]$, $x \geq x^3$. Revolving about the $x$-axis: outer radius $R = x$, inner radius $r = x^3$. Set up washer integral: $V = \pi\int_0^1 (R^2 - r^2)\,dx = \pi\int_0^1 (x^2 - x^6)\,dx$. Why distractors fail: Option B squares the difference $(x - x^3)^2$ instead of squaring each radius separately—this is a common error. Option C is the shell method about the $y$-axis, not the washer method about the $x$-axis. Option D reverses outer and inner radii, which would give a negative volume.

Answer: $x \geq x^2$ on $[0, 1]$, so the area is $\int_0^1 (x - x^2)\,dx$

Find intersection points: Set $x^2 = x$, so $x^2 - x = 0 \Rightarrow x(x-1) = 0$, giving $x = 0$ and $x = 1$. Determine which function is on top: For $0 < x < 1$, test $x = 0.5$: $f(0.5) = 0.5$ and $g(0.5) = 0.25$, so $x \geq x^2$ on this interval. Why the correct answer works: Option B correctly identifies $x$ as the upper curve and sets up $\int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$. Why distractors fail: Option A reverses the order, yielding a negative result. Option C falsely claims no intersection. Option D adds the functions instead of subtracting.

Answer: Yes; the shell height is $5 - (x^2 + 1) = 4 - x^2$, which is what was written

Determine the shell height: Each vertical shell at position $x$ extends from $y = x^2 + 1$ (bottom of the region) to $y = 5$ (top). The height is $5 - (x^2 + 1) = 4 - x^2$. Verify the student's expression: $5 - x^2 - 1 = 4 - x^2$, which matches. The integral $2\pi\int_0^2 x(4-x^2)\,dx$ is correct. Why distractors fail: Option B uses the wrong interval for height—measuring from $y = 0$ instead of $y = 5$. Option C ignores the lower boundary. Option D is false; the shell method works fine regardless of whether the region touches the axis.

Answer: $2\pi\int_{-2}^{2} (3 - x)(4 - x^2)\,dx$

Identify the shell components: With the axis of rotation at $x = 3$, the shell radius for a shell at position $x$ is $|3 - x| = 3 - x$ (since $-2 \leq x \leq 2 < 3$). The shell height is $4 - x^2$. Set up the integral: $V = 2\pi\int_{-2}^{2}(3 - x)(4 - x^2)\,dx$. Why distractors fail: Option A uses radius $x$, which would be correct for rotation about the $y$-axis, not $x = 3$. Option C incorrectly mixes washer and shell formulas. Option D uses $(x - 3)$, which is negative on $[-2,2]$, yielding a negative volume.

Answer: $\pi\int_0^1 [(e^x + 1)^2 - 1]\,dx$

Identify outer and inner radii: The axis is $y = -1$. The outer radius from $y = -1$ to $y = e^x$ is $R = e^x - (-1) = e^x + 1$. The inner radius from $y = -1$ to $y = 0$ is $r = 0 - (-1) = 1$. Set up the washer integral: $V = \pi\int_0^1 [R^2 - r^2]\,dx = \pi\int_0^1 [(e^x + 1)^2 - 1^2]\,dx = \pi\int_0^1 [(e^x + 1)^2 - 1]\,dx$. Why distractors fail: Option A ignores the shifted axis and omits the inner radius. Option C uses $e^x - 1$ as the outer radius, subtracting rather than adding the distance to the shifted axis. Option D uses the shell method formula structure with incorrect components.

Answer: The integral equals zero, but the actual area is $\int_0^{2\pi} |\sin(x)|\,dx = 4$

Signed area vs. geometric area: The integral $\int_0^{2\pi}\sin(x)\,dx = 0$ because the positive area on $[0, \pi]$ cancels with the negative area on $[\pi, 2\pi]$. However, geometric area requires $|\sin(x)|$. Compute the actual area: $\int_0^{2\pi}|\sin(x)|\,dx = \int_0^{\pi}\sin(x)\,dx + \int_{\pi}^{2\pi}(-\sin(x))\,dx = 2 + 2 = 4$. Why distractors fail: Option A confuses signed area with geometric area. Option C falsely concludes no area is enclosed. Option D incorrectly evaluates the integral.

Answer: No; the washer method requires $\pi\int[(\sqrt{x})^2 - (x/2)^2]\,dx$, squaring each radius separately

Common washer method error: A frequent mistake is computing $(R - r)^2$ instead of $R^2 - r^2$. The cross-sectional area of a washer is $\pi R^2 - \pi r^2$, not $\pi(R-r)^2$. Why the correct answer works: Option B correctly identifies the error: each radius must be squared independently before subtracting. $\pi\int_0^4 [x - x^2/4]\,dx$ is the correct setup. Why distractors fail: Option A validates the incorrect setup. Option C claims the shell method is required, which is not true—the washer method works fine with the correct formula. Option D has incorrect limits (the curves intersect at $x = 4$).

Answer: $\dfrac{3\pi}{4}$

Set up the disk method: Revolving about the $x$-axis: $V = \pi\int_1^4 (1/x)^2\,dx = \pi\int_1^4 x^{-2}\,dx$. Evaluate: $V = \pi\left[-x^{-1}\right]_1^4 = \pi\left(-\frac{1}{4} + 1\right) = \pi \cdot \frac{3}{4} = \frac{3\pi}{4}$. Why distractors fail: Option A ($\frac{\pi}{4}$) omits the constant from the lower limit. Option C ($\pi\ln 4$) would result from integrating $1/x$ (not $1/x^2$). Option D ($\frac{\pi}{2}$) is a miscalculation.