The base of a solid is the circle x^2 + y^2 = 9. Cross sections of the solid perpendicular to the x-axis are squares. What is the volume, in cubic units, of the solid?

A. 18
B. 36
C. 72
D. 144

Please help. Thank you in advance.

the solid is symmetric, so you just find the volume of the right side and double it.

Think of the solid as a stack of thin square sheets of thickness dx. The base of each square sheet has side 2y, where y = √(9-x^2), so its area is 4y^2=4(9-x^2)

Now just integrate to add up the volumes of all those thin squares.

so for x=3, volume= 4(27-9)=4(18)=72

actually, you forgot to double it. You found the volume of the right half only.

v = ∫[-3,3] 4(9-x^2) dx = 2∫[0,3] 4(9-x^2) dx = 144

Well, well, well, looks like we have a squared problem here! Let's pop some jokes and solve it together.

First, we take a look at the equation x^2 + y^2 = 9. Ah, a beautiful circle with a radius of 3. I bet it's good at going round and round!

Now, let's imagine slicing our solid perpendicular to the x-axis. Voila, we have squares! These squares will be facing upward, just like a square skydiving team.

Since we are dealing with a squared situation, we know that the area of each square will be the same, which means those squares can share a common side length. Let's call it "s".

But before we proceed, let's make sure our squares fit perfectly inside that circle. The longest possible diagonal of each square will be the diameter of the circle, which is twice the radius, or 6 units. Using Pythagoras, we find that s^2 + s^2 = (6)^2. Simplifying a bit, we get 2s^2 = 36, so s^2 = 18.

Now that we have the side length of our squares, it's time to calculate the volume of our solid. We will integrate, but no need to panic, integration can be fun too!

We will integrate along the x-axis from -3 to 3, as that's the range of our circle. The formula for volume is given by V = ∫(A(x)) dx, where A(x) is the area of the cross section at a given x.

Since each cross section is a square with side length s = √18, the area of each cross section is A(x) = (s)^2 = 18.

Now we put it all together and integrate:

V = ∫18 dx
= 18∫dx (from -3 to 3)
= 18 * (3 - (-3))
= 18 * 6
= 108

So, the volume of our solid is not on the menu, but it's definitely 108 cubic units!

Therefore, the correct answer is not directly given, but it is closest to option C which is 72. So, I guess option C is the clown's choice for today. Keep smiling, and happy math-ing!

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

First, let's visualize the problem. The base of the solid is a circle with radius 3, centered at the origin (0,0). Cross sections of the solid perpendicular to the x-axis are squares. This means that as we move along the x-axis from left to right, the height of each square cross section remains constant, while the width (side length) of the square changes proportionally.

To find the width (side length) of the square at each x-coordinate, let's consider a generic point on the base of the solid. The equation of the circle is x^2 + y^2 = 9. Solving for y, we get y = sqrt(9 - x^2). Since the square cross sections are perpendicular to the x-axis, the side length of a square at a particular x-coordinate is equal to 2 times the y-coordinate.

Therefore, the side length of the square at each x-coordinate is given by 2 * sqrt(9 - x^2).

Now, to find the volume of the solid, we construct an integral that represents the sum of all the square cross sections. We integrate the area formula for a square, A = s^2, where s is the side length, over the range of x-values that span the base of the solid.

The limits of integration will be from -3 to 3 since that covers the entire base of the solid.

The volume (V) of the solid is given by the integral:

V = ∫[from -3 to 3] (2 * sqrt(9 - x^2))^2 dx.

Simplifying the integral and performing the calculations yields:

V = ∫[from -3 to 3] 4(9 - x^2) dx
= 4 * (∫[from -3 to 3] 9 dx - ∫[from -3 to 3] x^2 dx)
= 4 * (9 * 2 - ∫[from -3 to 3] x^2 dx).

Integrating x^2 over the range -3 to 3, we get:

∫[from -3 to 3] x^2 dx = [x^3/3] from -3 to 3
= (3^3/3) - (-3^3/3)
= (27/3) - (-27/3)
= 27/3 + 27/3
= 54/3
= 18.

So, the volume of the solid is:

V = 4 * (9 * 2 - 18)
= 4 * (18 - 18)
= 4 * 0
= 0.

Therefore, the volume of the solid is 0 cubic units.

Based on the given answer choices, none of them matches the calculated volume. Please double-check the question or provide more information if needed.

Okay, so for the integral, I got 4(9x- 1/3x^3) + C.