Find the volume of the solid whose base is the semicircle y = √(25 - x²) where -5 ≤ x ≤ 5 and the cross sections perpendicular to the x-axis are squares.
Understand the Problem
The question is asking for the volume of a solid based on a semicircular base defined by the equation y = √(25 - x²) within the specified limits of x. The cross sections of this solid are squares perpendicular to the x-axis, indicating that the side length of each square correlates to the height of the semicircle at that x-value. We will calculate the area of each cross section and then integrate to find the total volume over the interval [-5, 5].
Answer
$$ V = \frac{500}{3} $$
Answer for screen readers
The volume of the solid is $$ V = \frac{500}{3} $$.
Steps to Solve
- Determine the side length of the square cross-section
The side length of the square cross-section at any point $x$ is equal to the height of the semicircle, which is given by the equation $y = \sqrt{25 - x^2}$. Therefore, the side length $s$ can be expressed as: $$ s = \sqrt{25 - x^2} $$
- Calculate the area of the square cross-section
The area $A$ of each square cross-section can be calculated using the formula for the area of a square, $A = s^2$. Substituting our expression for $s$ gives: $$ A = \left(\sqrt{25 - x^2}\right)^2 = 25 - x^2 $$
- Set up the integral for total volume
To find the total volume $V$ of the solid, we need to integrate the area of the cross-sections from $x = -5$ to $x = 5$: $$ V = \int_{-5}^{5} (25 - x^2) , dx $$
- Evaluate the integral
Now we will compute the integral: $$ V = \int_{-5}^{5} (25 - x^2) , dx $$
Split the integral: $$ V = \int_{-5}^{5} 25 , dx - \int_{-5}^{5} x^2 , dx $$
The first integral: $$ \int_{-5}^{5} 25 , dx = 25 \cdot [x]_{-5}^{5} = 25 \cdot (5 - (-5)) = 25 \cdot 10 = 250 $$
The second integral: $$ \int_{-5}^{5} x^2 , dx = \left[\frac{x^3}{3}\right]_{-5}^{5} = \frac{5^3}{3} - \frac{(-5)^3}{3} = \frac{125}{3} + \frac{125}{3} = \frac{250}{3} $$
- Final calculation of the volume
Now substituting back: $$ V = 250 - \frac{250}{3} $$
To subtract, we can express 250 as $\frac{750}{3}$: $$ V = \frac{750}{3} - \frac{250}{3} = \frac{500}{3} $$
The volume of the solid is $$ V = \frac{500}{3} $$.
More Information
The volume calculated represents the total space occupied by the solid with square cross-sections based on the semicircular base defined by $y = \sqrt{25 - x^2}$. This technique of using integration for volumes with variable cross-sections is common in calculus.
Tips
- Forgetting to square the side length when calculating the area of the cross-section. The area must be calculated as $A = s^2$, not just using $s$ directly.
- Not considering the limits of integration correctly; ensure the correct interval is used for the x-values.
- Missing the application of the properties of even functions when integrating. The function $25 - x^2$ is symmetric, and integrating from $-a$ to $a$ can simplify calculations.
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