9. Graphical Applications of Integrals

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?

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).

b. How is this formula changed if x0>b?

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = 3 ln x− x²/24 on [1, 6]

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = −8x−3 on [−2, 6] (Use calculus.)

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.