The base of a solid is bounded by y=2sqrtx, y=2 and x=4. Find the volume of solid if cross sections perpendicular to y=2 are semicircles
To find the volume of the solid, we first need to determine the shape of the cross-sections perpendicular to the line y=2. In this case, the cross-sections are semicircles.
Let's visualize the solid and its cross-sections. We have the curve y=2sqrt(x), the line y=2, and the line x=4. The region between y=2sqrt(x) and y=2 is bounded by x=4.
Now, since the cross-sections are semicircles, we can find the radius of each semicircle at a given y-coordinate by finding the difference between y=2 and the curve y=2sqrt(x). This difference represents the height of the semicircle.
The general formula to find the volume of a solid with semicircular cross-sections is:
V = ∫[a,b] A(y) dy
Where:
- V is the volume of the solid
- [a,b] is the interval over which we integrate
- A(y) is the area of the cross-section at each y-coordinate
- dy represents an infinitesimally small change in the y-coordinate
In our case, the interval of integration is [0,2] because the curve y=2sqrt(x) intersects y=2 at y=0 and y=2.
To find A(y), which is the area of the semicircle at each y-coordinate, we need to find the radius of the semicircle. As mentioned earlier, the radius is the difference between y=2 and y=2sqrt(x). So the radius is:
r = 2 - 2sqrt(x)
Since the area of a semicircle is (π * r²) / 2, the area of each semicircle at the y-coordinate is:
A(y) = (π * (2 - 2sqrt(x))²) / 2
Now we can substitute this A(y) into our volume formula and integrate:
V = ∫[0,2] (π * (2 - 2sqrt(x))²) / 2 dy
To evaluate this integral, we can use the Fundamental Theorem of Calculus or anti-differentiation. After integrating, we will have the volume of the solid.