BackDetermine whether the following statements are true and give an explanation or counterexample.
a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.
Determine whether the following statements are true and give an explanation or counterexample.
b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.
Determine whether the following statements are true and give an explanation or counterexample.
c. Let f(x)=12x^2. The area of the surface generated when the graph of f on [−4, 4] is revolved about the x-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the x-axis.
Determine whether the following statements are true and give an explanation or counterexample.
d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.
Consider the following curves on the given intervals.
b. Use a calculator or software to approximate the surface area.
y=cos x, for 0≤x≤π/2; about the x-axis
Consider the following curves on the given intervals.
a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis.
y=tan x , for 0≤x≤π/4; about the x-axis
Consider the following curves on the given intervals.
b. Use a calculator or software to approximate the surface area.
y=tan x , for 0≤x≤π/4; about the x-axis
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = x^3/2 / 3 − x^1/2 on [4, 16]
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = (x²+2)^3/2 / 3 on [0, 1]
Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).
A solid has a circular base; cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?
Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.
a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].
Why is the disk method a special case of the general slicing method?