Find the volume of a solid whose base is bounded by the parabola x=y^2 and the line x=9, having square cross-sections when sliced perpendicular to the x-axis.
To find the volume of the solid, we can use the method of cross-sections.
First, let's determine the limits of integration for the x-axis. Since the base of the solid is bounded by the parabola x=y^2 and the line x=9, we need to find the x-values where these two curves intersect.
Setting y^2 = 9 and solving for y, we get y = ±3. So the limits of integration for the y-axis are from -3 to 3.
Now, let's consider a square cross-section taken perpendicular to the x-axis at a given value of x.
For each value of x between 0 and 9, the y-coordinate of the parabola varies from -√x to √x. Therefore, the side length of the square cross-section is 2√x.
The volume of the square cross-section is given by V(x) = (side length)^2 = (2√x)^2 = 4x.
To find the volume of the entire solid, we integrate the volume function V(x) with respect to x over the interval [0, 9]:
V = ∫[0,9] 4x dx
Integrating 4x with respect to x gives V = 2x^2 evaluated from 0 to 9:
V = 2 * 9^2 - 2 * 0^2
V = 2 * 81 - 0
V = 162
So, the volume of the solid is 162 cubic units.
To find the volume of the solid, we need to integrate the areas of square cross-sections as we slice the solid perpendicular to the x-axis.
The base of the solid is bounded by the parabola x = y^2 and the line x = 9. Let's find the limits of integration for x.
Since the base is bounded by x = 9, our lower limit of integration is x = 0 and the upper limit is x = 9.
Now let's find the dimension of the square cross-sections at each value of x.
From the given information, we know that x = y^2. Solving for y, we have y = ±√x.
Since we want to find the volume using square cross-sections, the dimension of each square cross-section is equal to the side length.
Let's consider the slice at a general value of x. The side length of the square cross-section at x is equal to 2√x (since y = ±√x).
Now, let's integrate the area of the square cross-sections from x = 0 to x = 9.
The volume (V) of the solid is given by the integral:
V = ∫[0 to 9] (side length)^2 dx
V = ∫[0 to 9] (2√x)^2 dx
V = ∫[0 to 9] 4x dx
V = 4 ∫[0 to 9] x dx
Using the power rule of integration, we find:
V = 4 [x^2/2] [0 to 9]
V = 4 [(9^2/2) - (0^2/2)]
V = 4 (81/2)
V = 162 cubic units
Therefore, the volume of the solid is 162 cubic units.
the square has base 2y=2√x and height 2y=2√x and thickness dx
So,
v = ∫[0,9] (2√x)(2√x) dx
= ∫[0,9] 4x dx
...