To find the volume of the solid, we can use the method of slicing.

Problem Setup

• The base of the solid is a semicircle defined by $y = \sqrt{4 - x^2}$, where $-2 \leq x \leq 2$. This represents the upper half of a circle of radius 2 centered at the origin.
• The cross-sections perpendicular to the x-axis are squares.

Step 1: Area of Cross-Section

At each point $x$, the side length of the square cross-section is equal to $2y$ (since the diameter of the semicircle at that point is $2y$).

Thus, the area $A(x)$ of the square cross-section at a given $x$ is: $A(x) = \left( 2y \right)^2 = 4y^2 = 4\left( \sqrt{4 - x^2} \right)^2 = 4(4 - x^2)$ Simplifying this expression: $A(x) = 16 - 4x^2$

Step 2: Volume Integral

To find the volume of the solid, we integrate the area of the cross-section along the x-axis from $x = -2$ to $x = 2$. Therefore, the volume $V$ is given by: $V = \int_{-2}^{2} A(x) \, dx = \int_{-2}^{2} (16 - 4x^2) \, dx$

Step 3: Evaluate the Integral

We can now compute the integral:

$V = \int_{-2}^{2} 16 \, dx - \int_{-2}^{2} 4x^2 \, dx$

First, calculate each integral separately:

1. $\int_{-2}^{2} 16 \, dx = 16x \Big|_{-2}^{2} = 16(2) - 16(-2) = 32 + 32 = 64$
2. $\int_{-2}^{2} 4x^2 \, dx$ is an even function, so we can double the integral from 0 to 2: $\int_{-2}^{2} 4x^2 \, dx = 2 \int_0^2 4x^2 \, dx = 2 \times 4 \int_0^2 x^2 \, dx = 8 \left[ \frac{x^3}{3} \right]_0^2 = 8 \times \frac{8}{3} = \frac{64}{3}$

Now subtract: $V = 64 - \frac{64}{3} = \frac{192}{3} - \frac{64}{3} = \frac{128}{3}$

Thus, the volume of the solid is: $V = \frac{128}{3} \, \text{cubic units}$

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Would you like further explanation or details on any part of the process?

5 Related Questions:

1. How do I calculate the volume of a solid whose base is a different shape (e.g., a triangle)?
2. What is the difference between a square and a rectangular cross-section in volume problems?
3. Can this method of slicing be applied to irregular shapes? How?
4. How do you evaluate integrals that involve more complex functions, like trigonometric or exponential functions?
5. What other volume methods are useful for solids with cross-sectional shapes?

Tip:

For problems involving symmetric shapes, like this semicircle, it's often helpful to exploit symmetry to simplify the integral. In this case, recognizing that the integrand $4x^2$ is even allowed us to reduce the work by integrating from 0 to 2 and doubling the result.