ind the volume of the solid whose base is bounded by the graphs of y= x+1 and y= (x^2)+1, with the indicated cross sections taken perpendicular to the x-axis.
a) squares
b) rectangles of height 1
the answers are supposed to be
a. 81/10
b. 9/2
help with at least setup?
To find the volume of the solid, we need to integrate the area of the cross-sections taken perpendicular to the x-axis. Let's break down the problem into two parts:
Part a) Squares:
First, let's find the limits of integration. The base of the solid is bounded by the graphs of y = x + 1 and y = x^2 + 1. To find the limits, we need to set these two equations equal to each other and solve for x:
x + 1 = x^2 + 1
x^2 - x = 0
x(x - 1) = 0
So, the intersection points are x = 0 and x = 1. These are the limits of integration.
Now, we need to determine the side length of each square in terms of x. Since the squares are perpendicular to the x-axis, each square will have a side length equal to the difference of the two functions: (x^2 + 1) - (x + 1) = x^2 - x.
The area of each square is (side length)^2 = (x^2 - x)^2 = x^4 - 2x^3 + x^2.
To find the volume, we integrate the area expression from x = 0 to x = 1:
Volume = ∫[0,1] (x^4 - 2x^3 + x^2) dx
Evaluating this integral will give us the volume of the solid.
Part b) Rectangles of height 1:
For this case, the height of each rectangle is given as 1. The base of the solid remains the same, bounded by y = x + 1 and y = x^2 + 1.
Again, find the limits of integration by solving for the points where the two equations intersect: x = 0 and x = 1.
The width of each rectangle is given by the difference of the two functions: (x^2 + 1) - (x + 1) = x^2 - x.
The area of each rectangle is simply the product of its height and width, which gives us a constant area of 1 * (x^2 - x) = x^2 - x.
Integrating this area expression from x = 0 to x = 1 will give us the volume.
Volume = ∫[0,1] (x^2 - x) dx
Evaluating this integral will give us the volume of the solid.
Once you have set up the integrals, compute the integrals using integration techniques or a calculator to find the volumes.