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 = 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?
Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures).
c. Write an integral for the volume of the solid.
Let R be the region bounded by the curve y=cos^−1x and the x-axis on [0, 1]. A solid of revolution is obtained by revolving R about the y-axis (see figures).
b. Find an expression for the area A(y) of a cross section of the solid at a point y in [0,π/2].
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=4−x^2,x=2, and y=4; about the y-axis
Find the area of the surface generated when the given curve is revolved about the given axis.
y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis
Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.
Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)
b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy
Volumes without calculus Solve the following problems with and without calculus. A good picture helps.
b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?