1. Find the lateral (side) surface area of the cone generated by revolving the line segment. Lateral surface area 1 ba se circumference slant height 2
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1 s Section. Area of Surfaces of Revolution. Find the lateral (side) surface area of the cone generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height. Find the lateral surface area of the cone generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height. Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Frustum surface area r r slant height. Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Frustum surface area r r slant height. Find the area of the surface generated by y x, x, x axis 6. Find the area of the surface generated by y x x,. x., x axis 7. Find the area of the surface generated by y x, x, x axis 8. Find the area of the surface generated by x y y, y axis
2 . Find the area of the surface generated by x y y, y axis 8 y x, x ; y axis (Hint: Express. / evaluate the integral S y ds with appropriate limits.) ds in terms of, and. Did you know that if you can cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y r x shown here is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.)
3 Section. Area of Surfaces of Revolution Exercise Find the lateral (side) surface area of the cone generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height S b y a x x x ba se circumference r slant height Lateral surface area ba se circumference slant height
4 Find the lateral surface area of the cone generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Lateral surface area ba se circumference slant height x x y y x y x y d S x c y y y 8 ba se circumference 8 slant height Lateral surface area ba se circumference slant height 8 8
5 Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the x-axis. Check your answer with the geometry formula Frustum surface area r r slant height b S y a x x x x 6 r r slant height Frustum surface area r r slant height
6 Find the lateral surface area of the cone frustum generated by revolving the line segment y x, x, about the y-axis. Check your answer with the geometry formula Frustum surface area r r slant height y x y x x y S y y y y r r slant height Frustum surface area r r slant height
7 Find the area of the surface generated by x x x S x x x x 7 u / du 7 u / du y x, x, x axis u x du x du x x u x u / u / /
8 Find the area of the surface generated by y x, x, x axis / y x x x / x x x x x x S x x x x / u du x u u x du x u / u / /
9 Find the area of the surface generated by / y y y y y y y / y S y y / y / / y y d y / / x y y, y axis d y / / 8 / /
10 Find the area of the surface generated by y / y x y y, y axis 8 y y y y S y /8 y y /8 / u du /8 u / du /8 u y du y / /8 / /
11 / y x, x ; y axis (Hint: Express ds in terms of, and evaluate the integral S x ds with appropriate limits.) / x x x x ds x x x x x x x x S x ds x x u du x u u x du x x u u 8
12 Did you know that if you can cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y r x shown here is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.) y r x x x r x r x x r x r r x r r x ah S r x r a r x ah r a ah rx a r a h a rh
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