So I've got this graph here with three semicircles, and I need to evaluate the integral of the functions that represent these semicircles using the formula for circles. The semicircles are labeled with expressions above them: the smallest one has √(2x - x²), the medium one has √(-12 + 8x - x²), and the largest one has √(-72 + 18x - x²). I need to find the area under each semicircle and sum them up, I think. But maybe there's a better way.
First, I know that a full circle with radius r has the equation (x - h)² + (y - k)² = r², where (h, k) is the center of the circle. Since these are semicircles, their equations should be of the form y = √(r² - (x - h)²) + k, assuming they're the upper halves of the circles.
Looking at the smallest semicircle, it's labeled y = √(2x - x²). I can try to complete the square to find the center and radius.
2x - x² = - (x² - 2x) = - (x² - 2x + 1 - 1) = - [(x - 1)² - 1] = - (x - 1)² + 1.
So, y = √[1 - (x - 1)²]. That means it's a semicircle with center at (1, 0) and radius 1.
Similarly, for the medium semicircle, y = √(-12 + 8x - x²). Let's complete the square for -x² + 8x - 12.
- x² + 8x - 12 = - (x² - 8x + 12) = - [(x - 4)² - 16 + 12] = - (x - 4)² + 4.
So, y = √[4 - (x - 4)²] = 2 √[1 - ((x - 4)/2)²]. This is a semicircle with center at (4, 0) and radius 2.
For the largest semicircle, y = √(-72 + 18x - x²). Again, complete the square for -x² + 18x - 72.
- x² + 18x - 72 = - (x² - 18x + 72) = - [(x - 9)² - 81 + 72] = - (x - 9)² + 9.
Thus, y = √[9 - (x - 9)²] = 3 √[1 - ((x - 9)/3)²]. This is a semicircle with center at (9, 0) and radius 3.
Now, since all these semicircles are sitting on the x-axis, their areas would just be half the area of their full circles. So, the area of a full circle is πr², so the area of a semicircle is (1/2)πr².
So, for the smallest semicircle with r = 1, area = (1/2)π(1)² = π/2.
For the medium one with r = 2, area = (1/2)π(2)² = 2π.
For the largest one with r = 3, area = (1/2)π(3)² = (9/2)π.
Adding them up, total area = π/2 + 2π + (9/2)π = (1/2 + 4/2 + 9/2)π = (14/2)π = 7π.
But wait, maybe I need to evaluate the integrals explicitly. Let's see.
The integral of y = √(2x - x²) dx from x = a to x = b, where a and b are the points where the semicircle intersects the x-axis.
From y = √(2x - x²), set y = 0: 2x - x² = 0 ⇒ x(x - 2) = 0 ⇒ x = 0 or x = 2.
So, the integral is ∫ from 0 to 2 of √(2x - x²) dx.
Similarly, for y = √(-12 + 8x - x²), find where y = 0: -x² + 8x - 12 = 0 ⇒ x² - 8x + 12 = 0 ⇒ x = [8 ± √(64 - 48)]/2 = [8 ± √16]/2 = [8 ± 4]/2 ⇒ x = 2 or x = 6.
So, integral from 2 to 6 of √(-x² + 8x - 12) dx.
For y = √(-72 + 18x - x²), set y = 0: -x² + 18x - 72 = 0 ⇒ x² - 18x + 72 = 0 ⇒ x = [18 ± √(324 - 288)]/2 = [18 ± √36]/2 = [18 ± 6]/2 ⇒ x = 6 or x = 12.
So, integral from 6 to 12 of √(-x² + 18x - 72) dx.
Now, to evaluate these integrals, I can use the formula for the area of a semicircle, which is (1/2)πr², as I did earlier. Alternatively, I can evaluate each integral directly.
But since I already found the radii for each semicircle, it's easier to use the area formula.
So, the total area is π/2 + 2π + (9/2)π = 7π, as before.
Alternatively, if I have to evaluate the integrals directly, I can use substitution.
For example, for the first integral: ∫√(2x - x²) dx from 0 to 2.
From earlier, 2x - x² = 1 - (x - 1)².
Let u = x - 1, then when x = 0, u = -1; when x = 2, u = 1.
So, ∫√(1 - u²) du from u = -1 to u = 1.
This is the integral of √(1 - u²), which is a standard integral representing the area of a semicircle with radius 1.
The integral of √(1 - u²) du from -1 to 1 is π/2, which matches what I got earlier.
Similarly, for the second integral: ∫√(4 - (x - 4)²) dx from x = 2 to x = 6.
Let u = x - 4, then when x = 2, u = -2; when x = 6, u = 2.
So, ∫√(4 - u²) du from u = -2 to u = 2.
This is ∫2√(1 - (u/2)²) du from -2 to 2.
This is the integral of √(1 - (u/2)²), which is the area of a semicircle with radius 2, scaled by 2.
Alternatively, recognizing that ∫√(4 - u²) du is the area of a semicircle with radius 2, which is π(2)² / 2 = 2π.
Similarly, for the third integral: ∫√(9 - (x - 9)²) dx from x = 6 to x = 12.
Let u = x - 9, then when x = 6, u = -3; when x = 12, u = 3.
So, ∫√(9 - u²) du from u = -3 to u = 3.
This is ∫3√(1 - (u/3)²) du from -3 to 3.
This is the integral of √(1 - (u/3)²), which is the area of a semicircle with radius 3, which is (9/2)π.
Adding them up again, total area is π/2 + 2π + (9/2)π = 7π.
Therefore, the integral evaluates to 7π.
**Final Answer**
\[ \boxed{7\pi} \]

So I've got this problem here about finding the surface area bounded by two curves in what looks like polar coordinates. The curves are given by \( q = a \cdot \cos(\varphi) \) and \( p = b \cdot \cos(\varphi) \), with \( b > a > 0 \). There's a diagram accompanying it, but since I can't see it, I'll have to imagine it based on the description.
First off, I need to understand what these curves represent. Both equations are in terms of \( \varphi \), which is a common angular variable in polar coordinates. However, usually, in polar coordinates, you have \( r = f(\theta) \), where \( r \) is the radial distance and \( \theta \) is the angle. Here, we have \( q \) and \( p \) instead of \( r \) and \( \theta \), so I might need to adjust my notation accordingly or assume that \( q \) and \( p \) are analogous to \( r \) and \( \theta \).
Wait a minute, looking back at the user's message, it seems like there might be a typo or miscommunication in the variables. The diagram mentions \( \rho \) and \( \varphi \), but the equations use \( q \) and \( p \). This could be confusing. Maybe \( q \) is equivalent to \( \rho \), and \( \varphi \) is equivalent to \( \theta \). I'll assume that \( q \) is the radial variable and \( \varphi \) is the angular variable for the purpose of this problem.
So, rephrasing the curves:
- \( q = a \cdot \cos(\varphi) \)
- \( q = b \cdot \cos(\varphi) \), with \( b > a > 0 \)
I need to find the surface area bounded by these two curves.
First, I should visualize what these curves look like. Both are equations of the form \( q = k \cdot \cos(\varphi) \), which represent circles in polar coordinates. Specifically, \( q = a \cdot \cos(\varphi) \) is a circle with radius \( a/2 \) centered at \( (a/2, 0) \) in the \( q \)-\( \varphi \) plane, and similarly, \( q = b \cdot \cos(\varphi) \) is a circle with radius \( b/2 \) centered at \( (b/2, 0) \).
Since \( b > a > 0 \), the larger circle \( q = b \cdot \cos(\varphi) \) encompasses the smaller circle \( q = a \cdot \cos(\varphi) \), and they do not overlap except at the points where \( a \cdot \cos(\varphi) = b \cdot \cos(\varphi) \), which would only happen if \( a = b \) or \( \cos(\varphi) = 0 \). Since \( a \neq b \), the curves intersect where \( \cos(\varphi) = 0 \), which is at \( \varphi = \pi/2 \) and \( \varphi = 3\pi/2 \). At these points, \( q = 0 \), which corresponds to the origin.
So, the region bounded by these two curves is the area between these two circles, excluding the parts where \( q < 0 \), which isn't physically meaningful in polar coordinates.
Now, to find the surface area between these two curves, I need to integrate the difference between the two radial functions over the appropriate range of \( \varphi \).
The formula for the area between two polar curves is:
\[
A = \frac{1}{2} \int_{\varphi_1}^{\varphi_2} (r_{\text{outer}}^2 - r_{\text{inner}}^2) \, d\varphi
\]
In this case, \( r_{\text{outer}} = b \cdot \cos(\varphi) \) and \( r_{\text{inner}} = a \cdot \cos(\varphi) \), so:
\[
A = \frac{1}{2} \int_{\varphi_1}^{\varphi_2} \left( (b \cdot \cos(\varphi))^2 - (a \cdot \cos(\varphi))^2 \right) \, d\varphi
\]
Simplifying the integrand:
\[
(b^2 \cos^2(\varphi) - a^2 \cos^2(\varphi)) = (b^2 - a^2) \cos^2(\varphi)
\]
So, the integral becomes:
\[
A = \frac{1}{2} \int_{\varphi_1}^{\varphi_2} (b^2 - a^2) \cos^2(\varphi) \, d\varphi
\]
Now, I need to determine the limits of integration, \( \varphi_1 \) and \( \varphi_2 \). The region of interest is between the two circles, and since both curves are symmetric about the horizontal axis (where \( \varphi = 0 \)), I can integrate from \( \varphi = -\alpha \) to \( \varphi = \alpha \), where \( \alpha \) is the angle where the curves intersect, but since they only intersect at \( \varphi = \pi/2 \) and \( \varphi = 3\pi/2 \), where \( q = 0 \), I need to be careful.
Alternatively, I can integrate over the entire range where \( a \cdot \cos(\varphi) \leq q \leq b \cdot \cos(\varphi) \), which is from \( \varphi = -\pi/2 \) to \( \varphi = \pi/2 \), because outside this range, either \( q < 0 \), which isn't meaningful, or \( q = 0 \).
Wait, actually, in polar coordinates, \( q \) can be negative, but often we consider only \( q \geq 0 \). However, in some contexts, negative \( q \) can represent points in the opposite direction. But for the purpose of area, it's probably best to stick with \( q \geq 0 \).
Given that, the region of interest is from \( \varphi = -\pi/2 \) to \( \varphi = \pi/2 \), where both \( q = a \cdot \cos(\varphi) \) and \( q = b \cdot \cos(\varphi) \) are positive.
Therefore, the integral should be:
\[
A = \frac{1}{2} \int_{-\pi/2}^{\pi/2} (b^2 - a^2) \cos^2(\varphi) \, d\varphi
\]
I can factor out the constant \( (b^2 - a^2) \):
\[
A = \frac{1}{2} (b^2 - a^2) \int_{-\pi/2}^{\pi/2} \cos^2(\varphi) \, d\varphi
\]
Now, I need to evaluate the integral of \( \cos^2(\varphi) \) from \( -\pi/2 \) to \( \pi/2 \). I recall that \( \cos^2(\varphi) = \frac{1 + \cos(2\varphi)}{2} \), which might make the integration easier.
So, substituting:
\[
\int \cos^2(\varphi) \, d\varphi = \int \frac{1 + \cos(2\varphi)}{2} \, d\varphi = \frac{1}{2} \int (1 + \cos(2\varphi)) \, d\varphi = \frac{1}{2} \left( \varphi + \frac{\sin(2\varphi)}{2} \right) + C
\]
Now, evaluating this from \( -\pi/2 \) to \( \pi/2 \):
\[
\left. \frac{1}{2} \left( \varphi + \frac{\sin(2\varphi)}{2} \right) \right|_{-\pi/2}^{\pi/2} = \frac{1}{2} \left( \left( \frac{\pi}{2} + \frac{\sin(\pi)}{2} \right) - \left( -\frac{\pi}{2} + \frac{\sin(-\pi)}{2} \right) \right)
\]
Since \( \sin(\pi) = 0 \) and \( \sin(-\pi) = 0 \), this simplifies to:
\[
\frac{1}{2} \left( \frac{\pi}{2} + \frac{\pi}{2} \right) = \frac{1}{2} \cdot \pi = \frac{\pi}{2}
\]
Therefore, the area is:
\[
A = \frac{1}{2} (b^2 - a^2) \cdot \frac{\pi}{2} = \frac{\pi}{4} (b^2 - a^2)
\]
But wait a second, this seems too straightforward, and I'm wondering if I've missed something. Let me double-check.
First, the integral of \( \cos^2(\varphi) \) over \( -\pi/2 \) to \( \pi/2 \) indeed should be \( \pi/2 \), because \( \cos^2(\varphi) \) is symmetric and integrates to \( \pi \) over a full period, so over half a period, it should be \( \pi/2 \).
Moreover, the factor of \( 1/2 \) in front of the integral is correct for the area between two polar curves.
However, I recall that the area of a circle in polar coordinates is \( \pi r^2 \), and here, the area between two circles with radii \( a/2 \) and \( b/2 \) should be \( \pi (b/2)^2 - \pi (a/2)^2 = \pi (b^2 - a^2)/4 \), which matches my result.
But let's consider another approach to confirm.
Alternatively, I can think of the area as the difference between the areas of the two circles:
\[
A = \text{area of larger circle} - \text{area of smaller circle}
\]
But wait, the curves are not exactly circles centered at the origin. They are circles offset from the origin along the horizontal axis.
Specifically, \( q = a \cdot \cos(\varphi) \) can be rewritten in Cartesian coordinates (assuming \( q \) is the radial distance and \( \varphi \) is the angle) as:
\[
q = a \cdot \cos(\varphi) \implies q^2 = a q \cos(\varphi) \implies x^2 + y^2 = a x
\]
Completing the square:
\[
x^2 - a x + y^2 = 0 \implies \left( x - \frac{a}{2} \right)^2 - \left( \frac{a}{2} \right)^2 + y^2 = 0 \implies \left( x - \frac{a}{2} \right)^2 + y^2 = \left( \frac{a}{2} \right)^2
\]
So, it's a circle with center at \( (a/2, 0) \) and radius \( a/2 \).
Similarly, \( q = b \cdot \cos(\varphi) \) is a circle with center at \( (b/2, 0) \) and radius \( b/2 \).
Given that, the area between these two circles would indeed be \( \pi (b/2)^2 - \pi (a/2)^2 = \pi (b^2 - a^2)/4 \), which matches my earlier result.
Therefore, I'm confident that the surface area bounded by the curves \( q = a \cdot \cos(\varphi) \) and \( q = b \cdot \cos(\varphi) \), with \( b > a > 0 \), is \( \frac{\pi}{4} (b^2 - a^2) \).
**Final Answer**
\[ \boxed{ \frac{\pi}{4} (b^{2} - a^{2}) } \]

5. 如图甲所示，理想变压器原线圈接在正弦式交流电源上，输入电压 u 随时间 t 变化的图像如图乙所示，副线圈接规格为“6V，3W”的灯泡。若灯泡正常发光，下列说法正确的是（ ）
A. 原线圈两端电压的有效值为 \(24\sqrt{2}\) V
B. 副线圈中电流的有效值为 0.5A
C. 原、副线圈匝数之比为 1：4
首先，我需要理解这个题目。题目中有一个理想变压器，原线圈接在正弦式交流电源上，输入电压随时间的变化图像给出，副线圈接了一个规格为“6V，3W”的灯泡，并且这个灯泡正常发光。我需要根据这些信息来判断几个选项的正误。
首先，来看选项A，原线圈两端电压的有效值是多少。从图乙中，我看到输入电压u随时间t变化的图像，最大电压是\(24\sqrt{2}\) V。因为是正弦式交流电，所以有效值应该是最大值除以根号2，也就是 \(24\sqrt{2} / \sqrt{2} = 24\) V。所以，原线圈两端电压的有效值是24V，而不是 \(24\sqrt{2}\) V。所以选项A是错误的。
接下来是选项B，副线圈中电流的有效值是多少。题目中说灯泡正常发光，它的规格是“6V，3W”，这意味着灯泡在额定电压下工作，消耗的功率是3W。根据功率公式 P = UI，可以求出电流 I = P / U = 3W / 6V = 0.5A。所以，副线圈中的电流有效值是0.5A，选项B是正确的。
最后是选项C，原、副线圈匝数之比是多少。理想变压器的电压比等于匝数比，即 \(n_1 / n_2 = U_1 / U_2\)。已知原线圈电压有效值 U1 = 24V，副线圈电压有效值 U2 = 6V（因为灯泡正常发光，所以副线圈电压等于灯泡额定电压）。所以，匝数比 \(n_1 / n_2 = 24 / 6 = 4 / 1\)，也就是4：1，不是1：4。所以选项C是错误的。
综上所述，只有选项B是正确的。
不过，我再仔细检查一下，确保没有遗漏。首先，原线圈电压有效值确实是24V，这个没问题。副线圈电流是0.5A，这个也是对的。匝数比是4：1，不是1：4，所以C是错的。看来我的判断是对的，只有B是正确的。
嗯，再看看有没有其他需要考虑的地方。比如，变压器的功率是否守恒，也就是输入功率等于输出功率。副线圈负载是3W，那么原线圈的输入功率也应该 是3W。根据 P = UI，原线圈电流 I1 = P / U1 = 3W / 24V = 0.125A。这个信息虽然题目没问，但可以用来验证能量守恒。
总之，我认为选项B是正确的，而A和C是错误的。所以答案应该是B。
**答案**
B

5. 如图甲所示，理想变压器原线圈接在正弦式交流电源上，输入电压 u 随时间 t 变化的图像如图乙所示，副线圈接规格为“6V，3W”的灯泡。若灯泡正常发光，下列说法正确的是（ ）
A. 原线圈两端电压的有效值为 \(24\sqrt{2}\) V
B. 副线圈中电流的有效值为 0.5A
C. 原、副线圈匝数之比为 1：4
首先，我需要理解这个题目。题目中有一个理想变压器，原线圈接在正弦式交流电源上，输入电压随时间的变化图像给出，副线圈接了一个规格为“6V，3W”的灯泡，并且这个灯泡正常发光。我需要根据这些信息来判断几个选项的正误。
首先，来看选项A，原线圈两端电压的有效值是多少。从图乙中，我看到输入电压u随时间t变化的图像，最大电压是\(24\sqrt{2}\) V。因为是正弦式交流电，所以有效值应该是最大值除以根号2，也就是 \(24\sqrt{2} / \sqrt{2} = 24\) V。所以，原线圈两端电压的有效值是24V，而不是 \(24\sqrt{2}\) V。所以选项A是错误的。
接下来是选项B，副线圈中电流的有效值是多少。题目中说灯泡正常发光，它的规格是“6V，3W”，这意味着灯泡在额定电压下工作，消耗的功率是3W。根据功率公式 P = UI，可以求出电流 I = P / U = 3W / 6V = 0.5A。所以，副线圈中的电流有效值是0.5A，选项B是正确的。
最后是选项C，原、副线圈匝数之比是多少。理想变压器的电压比等于匝数比，即 \(n_1 / n_2 = U_1 / U_2\)。已知原线圈电压有效值 U1 = 24V，副线圈电压有效值 U2 = 6V（因为灯泡正常发光，所以副线圈电压等于灯泡额定电压）。所以，匝数比 \(n_1 / n_2 = 24 / 6 = 4 / 1\)，也就是4：1，不是1：4。所以选项C是错误的。
综上所述，只有选项B是正确的。
不过，我再仔细检查一下，确保没有遗漏。首先，原线圈电压有效值确实是24V，这个没问题。副线圈电流是0.5A，这个也是对的。匝数比是4：1，不是1：4，所以C是错的。看来我的判断是对的，只有B是正确的。
嗯，再看看有没有其他需要考虑的地方。比如，变压器的功率是否守恒，也就是输入功率等于输出功率。副线圈负载是3W，那么原线圈的输入功率也应该 是3W。根据 P = UI，原线圈电流 I1 = P / U1 = 3W / 24V = 0.125A。这个信息虽然题目没问，但可以用来验证能量守恒。
总之，我认为选项B是正确的，而A和C是错误的。所以答案应该是B。
**答案**
B

少年宫在图书馆的什么方向？
首先，我需要理解这个问题的背景。应该是一个地图或者是一个方位图，上面有各种建筑，比如图书馆、少年宫、学校等等。问题是在问少年宫相对于图书馆的方向。
既然有地图，那肯定有方向指示，比如通常地图上北向上，东向右，南向下，西向左。但是，为了更准确，我应该看看地图上是否有指南针或者方向标。
在题目中，我看到有“北”和“东”的方向标。北向上，东向右，那么南就向下，西向左。
现在，我需要找到图书馆和少年宫的位置。
从图上看，图书馆应该是在左边，有一个房子的图标，上面写着“图书馆”。
然后，少年宫也在左边，有一个类似宫殿的图标，上面写着“少年宫”。
看起来，少年宫和图书馆是在同一侧，都是在地图的左边。
我需要确定它们之间的相对位置。
从图上来看，图书馆似乎在上面一点，少年宫在下面一点。
但是，要确定方向，我需要根据地图的坐标系来判断。
假设地图是按照常规的北向上，东向右，南向下，西向左来绘制的。
那么，如果图书馆在上面，少年宫在下面，那么少年宫应该是在图书馆的南方。
但是，我需要确认一下地图的布局。
从图上看到，图书馆在左上角，少年宫在左下角，学校在中间，体育馆在上边中间，动物园在右上角，医院在左中间，邮局在右中间，电影院在右下角，商场在下边中间。
所以，根据这个布局，少年宫在图书馆的下方，也就是南方。
但是，可能还有其他的方向需要考虑，比如东南、西南等等。
我需要更精确地判断它们之间的方向关系。
从图上来看，图书馆和少年宫之间似乎有一条竖线，可能代表道路或者其他分界线。
如果图书馆在上面，少年宫在下面，那么少年宫应该是在图书馆的正南方。
但是，可能由于地图的比例尺或者建筑物的实际位置，可能会有偏差。
我需要根据地图上的比例尺或者距离来判断，但是从题目中没有给出具体的比例尺，所以可能只能根据图上的位置来大致判断。
另外，可能需要考虑实际的地理位置，比如这些地方可能在中国的某个城市，但是没有具体的信息。
所以，基于地图的布局，我判断少年宫在图书馆的正南方。
不过，为了确保答案的准确性，我应该再检查一下。
首先，确认方向标，北向上，东向右，南向下，西向左。
图书馆在左上角，少年宫在左下角，所以应该是南方。
如果地图有更详细的坐标或者距离信息，可能会更准确，但是目前只能根据位置来判断。
所以，最终的答案应该是少年宫在图书馆的正南方。
**答案**
少年宫在图书馆的正南方

So I have this L-shaped sofa with dimensions provided, and I need to find its volume. The measurements are as follows:
- Length: 300 cm
- Width (both sections): 90 cm
- Depth (chaise section): 180 cm
First, I need to understand the shape and how it affects the volume calculation. An L-shaped sofa consists of two main parts: the straight section and the chaise lounge section.
Let me visualize this:
- The straight section is 300 cm in length and 90 cm in depth.
- The chaise section extends from the straight section, adding another 180 cm in depth and 90 cm in width.
To find the volume, I should consider the sofa as a combination of rectangular prisms.
Perhaps I can divide the sofa into two parts:
1. The straight section: a rectangular prism with dimensions 300 cm x 90 cm x 90 cm.
2. The chaise section: another rectangular prism with dimensions 180 cm x 90 cm x 90 cm.
Wait, but when I look at the dimensions, the chaise section is 180 cm in depth and 90 cm in width. However, since it's an L-shape, the width of the chaise section might overlap with the straight section.
Let me double-check the dimensions.
Looking at the image:
- The overall length is 300 cm.
- The width is 90 cm for both sections.
- The depth of the chaise is 180 cm.
I think the straight section is 300 cm in length and 90 cm in width, and the chaise section is 180 cm in depth and 90 cm in width, attached to the straight section.
In this case, the volume would be the sum of the volumes of these two sections.
First, calculate the volume of the straight section:
Volume_straight = length x width x height
Here, length = 300 cm, width = 90 cm, and height = 90 cm.
Volume_straight = 300 cm x 90 cm x 90 cm
Wait, but I need to confirm if the height is indeed 90 cm. The image shows a vertical dimension of 90 cm, which could be the height.
So, Volume_straight = 300 cm x 90 cm x 90 cm
Similarly, for the chaise section:
Volume_chaise = depth x width x height
Depth = 180 cm, width = 90 cm, height = 90 cm.
Volume_chaise = 180 cm x 90 cm x 90 cm
Then, the total volume would be Volume_total = Volume_straight + Volume_chaise
Let me calculate that.
First, Volume_straight = 300 cm x 90 cm x 90 cm = 2,430,000 cubic cm
Second, Volume_chaise = 180 cm x 90 cm x 90 cm = 1,458,000 cubic cm
Total volume = 2,430,000 + 1,458,000 = 3,888,000 cubic cm
But, I should consider if there's any overlap or shared volume between the two sections.
Looking back, in an L-shape, the chaise section shares some space with the straight section, so perhaps simply adding the volumes isn't accurate.
Maybe I should think of the entire sofa as a single rectangular prism with length 300 cm, width 180 cm (since the chaise adds 180 cm to the width), and height 90 cm.
Wait, but the width of the straight section is 90 cm, and the chaise adds another 90 cm to the width, making the total width 180 cm.
So, total dimensions would be:
Length: 300 cm
Width: 180 cm
Height: 90 cm
Therefore, total volume = length x width x height = 300 cm x 180 cm x 90 cm = 4,860,000 cubic cm
However, this seems larger than the previous calculation. Which one is correct?
Let me think again.
Option 1: Sum of two separate volumes:
Volume_straight + Volume_chaise = 2,430,000 + 1,458,000 = 3,888,000 cubic cm
Option 2: Entire sofa as one big rectangular prism:
Volume_total = 300 cm x 180 cm x 90 cm = 4,860,000 cubic cm
But this seems incorrect because it's overcounting the shared space where the two sections meet.
So, Option 1 is more accurate.
Alternatively, perhaps I need to consider the chaise section as an extension that doesn't add its full volume but only the additional volume it provides.
Wait, maybe I should subtract the overlapping volume.
Let me try another approach.
Consider the straight section as a rectangular prism: 300 cm x 90 cm x 90 cm.
The chaise section extends 180 cm in depth and 90 cm in width, but since it's attached to the straight section, there is an overlap in width.
Actually, in an L-shape, the chaise section shares the width with the straight section, so the additional volume added by the chaise is only the part that doesn't overlap.
Wait, this is getting complicated.
Maybe I should look at it differently.
Let's consider the entire base area first.
The base of the sofa would be the area in contact with the floor.
The straight section is 300 cm in length and 90 cm in width.
The chaise section adds another 180 cm in depth, making the total width 90 cm + 180 cm = 270 cm.
Wait, no, if the straight section is 300 cm in length and 90 cm in width, and the chaise adds 180 cm in depth, then the base area would be a rectangle of 300 cm x 180 cm.
But that doesn't account for the L-shape properly.
Alternatively, perhaps the base is L-shaped, with one side 300 cm and the other 180 cm, intersecting at a corner.
In that case, the total base area would be:
Area_base = (300 cm x 90 cm) + (180 cm x 90 cm) - (overlap area)
But what is the overlap area?
If the chaise section is attached to the straight section, the overlap would be the area where they connect.
Assuming they connect at a 90-degree angle, the overlap would be a square of side 90 cm.
Therefore, Area_overlap = 90 cm x 90 cm = 8,100 square cm
Thus, Area_base = (300 cm x 90 cm) + (180 cm x 90 cm) - 8,100 square cm
Calculate that:
(27,000 square cm) + (16,200 square cm) = 43,200 square cm
Minus 8,100 square cm = 35,100 square cm
Then, the height is 90 cm, so Volume_total = Area_base x height = 35,100 square cm x 90 cm = 3,159,000 cubic cm
This is different from both previous calculations.
Which one is correct?
I think this method accounts for the overlap properly.
Alternatively, perhaps I can use integration to find the volume, but that might be overkill for this problem.
Wait, another way: divide the sofa into smaller rectangular prisms whose volumes can be added up accurately.
Let's try that.
Divide the sofa into three parts:
1. The main seat section: 300 cm x 90 cm x 90 cm
2. The chaise section: 180 cm x 90 cm x 90 cm
3. Subtract the overlapping part: 90 cm x 90 cm x 90 cm
Then, Volume_total = (300 cm x 90 cm x 90 cm) + (180 cm x 90 cm x 90 cm) - (90 cm x 90 cm x 90 cm)
Calculate each part:
Volume_main = 300 cm x 90 cm x 90 cm = 2,430,000 cubic cm
Volume_chaise = 180 cm x 90 cm x 90 cm = 1,458,000 cubic cm
Volume_overlap = 90 cm x 90 cm x 90 cm = 729,000 cubic cm
Therefore, Volume_total = 2,430,000 + 1,458,000 - 729,000 = 3,159,000 cubic cm
This matches the previous calculation using the base area method.
So, the correct volume is 3,159,000 cubic cm.
But wait, earlier I thought the total base area was 35,100 square cm multiplied by 90 cm, which gives 3,159,000 cubic cm.
Yes, that seems consistent.
Therefore, the volume of the sofa is 3,159,000 cubic cm.
**Final Answer**
\[ \boxed{3159000\ \text{cubic cm}} \]