Consider the functions f(x) = (x^3/x^4+1) and g(x) = (x/x^4+1). Let R denote the region in the first x4+1 x4+1 quadrant bounded by the curves y = f(x) and y = g(x). Find the exact volume of the solid that has R as its base if every cross section by a plane perpendicular to the x-axis is a rectangle of height 3. ("Exact volume" means no calculator numbers.)
To find the exact volume of the solid, we need to integrate the area of each cross section with respect to x.
First, let's find the points where the curves intersect. To do this, we set f(x) equal to g(x) and solve for x:
f(x) = g(x)
x^3/(x^4+1) = x/(x^4+1)
Cross multiplying:
x^3 = x^2
Dividing both sides by x^2:
x = 1
So the curves intersect at x = 1.
Now, let's determine the limits of integration. Since we are only interested in the first quadrant, the x-values range from 0 to 1.
The height of each rectangle is given as 3, so the area of each cross section is 3 times the difference between the curve functions.
A(x) = 3 * (f(x) - g(x))
Substituting the functions given:
A(x) = 3 * [(x^3/(x^4+1)) - (x/(x^4+1))]
We can simplify this expression using a common denominator:
A(x) = 3 * [(x^3 - x)/(x^4 + 1)]
Now, integrating A(x) with respect to x will give us the volume of the solid:
V = ∫[0 to 1] A(x) dx
= ∫[0 to 1] 3 * [(x^3 - x)/(x^4 + 1)] dx
To evaluate this integral, we can use techniques like u-substitution or partial fraction decomposition.
Once the integral is evaluated, we will have the exact volume of the solid.