9. Graphical Applications of Integrals

Volumes

Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = π and y = 2 about the line y = 2.

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

b. the line y = ―1

Volumes

Find the volume of the solid generated by revolving the region between the x-axis and curve y = x² ―2x about

d. the line y = 2

Volumes

Volume of a solid sphere hole A round hole of radius √3 ft is bored through the center of a solid sphere of radius 2 ft. Find the volume of material removed from the sphere.

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

y = sin x, 0 ≤ x ≤ π

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

y = x²/4, 0 ≤ x ≤ 2

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

Finding surface area

Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis.

Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_____

y = √2x + 1 , 0 ≤ x ≤ 3 ; x-axis"

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_______

y = √4y ― y² , 1 ≤ y ≤ 2 ; y-axis

80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.

Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.