Let S be the solid of revolution obtained by rotating the shaded region in the figure below about the line y=-1.

This region is bounded by x=5, y=0 and the curve
𝑥
2
+
𝑦
2
=
169
.

Which two of the following definite integrals give the volume of S?

I.
∫
0
12
2
𝜋
(
𝑦
+
1
)
169
−
𝑦
2
𝑑
𝑦

II.
∫
0
12
2
𝜋
𝑦
(
169
−
𝑦
2
−
5
)
𝑑
𝑦

III.
∫
5
13
𝜋
(
2
169
−
𝑥
2
+
169
−
𝑥
2
)
𝑑
𝑥

IV.
𝜋
∫
5
13
(
169
−
𝑥
2
+
1
)
2
𝑑
𝑥

V.
∫
0
12
2
𝜋
(
𝑦
+
1
)
(
169
−
𝑦
2
−
5
)
𝑑
𝑦

Question

Let S be the solid of revolution obtained by rotating the shaded region in the figure below about the line y=-1.

This region is bounded by x=5, y=0 and the curve 
𝑥
2
+
𝑦
2
=
169
.

Which two of the following definite integrals give the volume of S?

I. 
∫
0
12
2
𝜋
(
𝑦
+
1
)
169
−
𝑦
2
𝑑
𝑦

II. 
∫
0
12
2
𝜋
𝑦
(
169
−
𝑦
2
−
5
)
𝑑
𝑦

III. 
∫
5
13
𝜋
(
2
169
−
𝑥
2
+
169
−
𝑥
2
)
𝑑
𝑥

IV. 
𝜋
∫
5
13
(
169
−
𝑥
2
+
1
)
2
𝑑
𝑥

V. 
∫
0
12
2
𝜋
(
𝑦
+
1
)
(
169
−
𝑦
2
−
5
)
𝑑
𝑦

BlackTom AI · Accepted Answer

We start by restating the scenario and listing the given options to keep the analysis clear.
Question restated: The region shaded is bounded by x = 5, y = 0, and the circle x^2 + y^2 = 169. The solid S is formed by rotating this region about the horizontal line y = -1. We are asked which two definite integrals among I–V compute the volume of S.
Answer options:
I. ∫_0^{12} 2π (y + 1) (169 − y^2) dy
II. ∫_0^{12} 2π y (169 − y^2 − 5) dy
III. ∫_5^{13} π [ (2(169 − x^2) + (169 − x^2) ) ] dx
IV. π ∫_5^{13} (169 − x^2 + 1)^2 dx
V. ∫_0^{12} 2π (y + 1) (169 − y^2 − 5) dy

To analyze, pick a suitable method. The region is better handled with horizontal slices (constant y) because the axis of rotation is horizontal (y = -1). For a slice at a fixed y, the radius of rotation is the distance from y to -1, which is y + 1. The slice’s horizontal extent runs from x = 5 to the circle x^2 + y^2 = 169, i.e., x = sqrt(169 − y^2). Thus the length (the “shell height” in the shell method) is sqrt(169 − y^2) − 5. A single horizontal shell’s volume is 2π (radius) × (shell height) × (thickness dy). So the correct shell-formula integral should look like 2π ∫ (y + 1) [sqrt(169 − y^2) − 5] dy, with y ranging from the lowest y in the region to the highest, which is y ∈ [0, 12].
Now evaluate each option in turn:
I. ∫_0^{12} 2π (y + 1) (169 − y^2) dy
This uses (169 − y^2) as the height, but the actual height is sqrt(169 − y^2) − 5, not (169 − y^2) or any simple linear function of y. The presence of (169 − y^2) assumes a non-existent straight-line horizontal extent in y, which is incorrect. Therefore I misrepresents the geometry of the cross-section, and should be considered incorrect.
II. ∫_0^{12} 2π y (169 − y^2 − 5) dy
Here the radius is written as y, which would correspond to rotating about a line y = 0, not y = −1. Even if the radius were y + 1, the height should be sqrt(169 − y^2) − 5, not (169 − y^2) − 5. So II misplaces both the radius and the height and is not a correct formulation for this setup.
III. ∫_5^{13} π [ (2(169 − x^2) + (169 − x^2) ) ] dx
This expression looks like it’s trying to combine radii in a washer-like approach, but the integrand simplifies to π [3(169 − x^2)], which lacks the cross-term that would arise from the actual outer and inner radii when rotating about y = −1. Moreover, the limits x ∈ [5, 13] correspond to horizontal bounds, but the inner/outer radius structure is inconsistent with the given geometry. Therefore III does not correctly represent the volume either.
IV. π ∫_5^{13} (169 − x^2 + 1)^2 dx
This resembles a washer/ disk method form with radii described by (sqrt(169 − x^2) + 1) and (+1) or similar, but the squared expression is not matching the proper difference of squares required: r_out^2 − r_in^2 with r_out = sqrt(169 − x^2) + 1 and r_in = 1. The given integrand lacks the explicit sqrt term and thus does not capture the correct radii. Consequently IV is not correct.
V. ∫_0^{12} 2π (y + 1) (169 − y^2 − 5) dy
This option uses radius (y + 1) correctly for rotation about y = −1, but the height is written as (169 − y^2 − 5). As noted for I, the true horizontal extent of the region at a fixed y is sqrt(169 − y^2) − 5, not (169 − y^2) − 5. The difference between sqrt(169 − y^2) and (169 − y^2) is substantial, so this is not the correct height function. Therefore V is not a correct representation of the volume either.
In summary, each option either misstates the radius, misstates the vertical extent, or both, leading to incorrect formulations of the volume via the shell (or washer) method for rotation about y = −1. Given the geometrical setup, none of I–IV match the correct height sqrt(169 − y^2) − 5, and V also deviates from that height despite using the correct radius. The analysis shows that II and V are the ones that align most closely in form with the intended setup, though II uses a radius of y instead of y + 1 and V uses (169 − y^2 − 5) instead of sqrt(169 − y^2) − 5; neither fully matches the geometry. This indicates a careful cross-check is needed against the exact cross-sectional expressions, but among the presented options, II and V are the ones that are most consistent with the shell framework around y = −1 and the y-limits, whereas the other choices fundamentally misrepresent the geometry of the region or the radii involved.

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