a) To find the area of region R, we need to find the intersection points of the two graphs and then integrate over the interval between those points.
Step 1: Setting the two equations equal to each other.
sin(Ï€x) = x^3 - 4
Step 2: Solve for x.
This equation is transcendental, meaning it cannot be solved algebraically. We will need to use numerical methods, such as graphical or computational methods, to find the intersection points. One common approach is to use a graphing calculator or computer software to graph both equations and find their intersection points.
Step 3: Once we have found the intersection points, we can set up the integral to find the area.
The area can be calculated as the integral of the difference of the two functions over the interval where they intersect:
Area = ∫[a,b] (f(x) - g(x)) dx, where f(x) is the top function (sin(πx)) and g(x) is the bottom function (x^3 - 4).
b) To find the area of the part of region R below the horizontal line y = -2, we need to find the x-values where the graph of y = -2 intersects the region.
Step 1: Set the equation of the horizontal line equal to the bottom function.
-2 = x^3 - 4
Step 2: Solve for x.
Again, this equation is transcendental and requires numerical methods to find the x-values where the line intersects the curve.
Step 3: Set up the integral expression for the area below the horizontal line.
The integral expression will be similar to part a), but the limits of integration will be different:
Area = ∫[c,d] (-2 - g(x)) dx, where g(x) is the bottom function (x^3 - 4) and c, d are the x-values where the horizontal line intersects the curve.
c) To find the volume of the solid with R as its base and square cross-sections perpendicular to the x-axis, we need to calculate the area of each cross-section and integrate over the range of x.
Step 1: Determine the length of each side of the square cross-section.
The length of each side of the square cross-section is the difference between the top and bottom functions for a given x.
Step 2: Calculate the area of each cross-section.
The area of a square is the square of the side length, so the area of each cross-section is (f(x) - g(x))^2, where f(x) is the top function (sin(Ï€x)) and g(x) is the bottom function (x^3 - 4).
Step 3: Set up the integral expression for the volume.
The volume can be calculated as the integral of the area of each cross-section over the range of x:
Volume = ∫[a,b] (f(x) - g(x))^2 dx.
d) To find the volume of water in the pond, we need to calculate the volume of the region R when the depth of the water is given by h(x) = 3 - x.
Step 1: Determine the height of each cross-section.
The height of each cross-section is the value of h(x) for a given x.
Step 2: Calculate the area of each cross-section.
The area of each cross-section is the same as in part c), which is (f(x) - g(x))^2.
Step 3: Set up the integral expression for the volume of water.
The volume of water can be calculated as the integral of the area of each cross-section multiplied by the height over the range of x:
Volume of Water = ∫[a,b] h(x) * (f(x) - g(x))^2 dx.