Volume of Known Cross Sections

Volume of known cross sections

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1) In the figure, R is the shaded region in the first quadrant bounded by the

graph of y = 4ln(3 - x), the horizontal line y = 6, and the vertical line x = 2.

a) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is a square. Find the volume of the solid.

b) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is an equilateral triangle. Find the volume of the

solid.

2) Let T be the region in the first quadrant bounded by the graphs of and. The region R is the base of a

solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Find the volume of this solid.

3) Let R be the region bounded by the graphs of y = sin(πx) and y = x3 - 4x, as shown

in the figure.

a) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is an isosceles triangle. Find the volume of this

solid.

b) The region R models the surface of a small pond. At all points in R at a

distance x from the y-axis, the depth of the water is given by h(x) = 3 - x.

Find the volume of water in the pond.

4) Region T is bounded by and . Cross-sections perpendicular to the y-axis are

rectangles whose height is four times its length in base T. Find the volume of the solid.

5) Region S is bounded by the ellipse. Cross sections perpendicular to the x-axis are isosceles

right triangles with hypotenuse in the xy-plane. Find the volume of the solid.

Volume of known cross sections

Show all appropriate integrals! Calculator active.

1) In the figure, R is the shaded region in the first quadrant bounded by the

graph of y = 4ln(3 - x), the horizontal line y = 6, and the vertical line x = 2.

a) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is a square. Find the volume of the solid.

b) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is an equilateral triangle. Find the volume of the

solid.

2) Let T be the region in the first quadrant bounded by the graphs of and. The region R is the base of a

solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Find the volume of this solid.

3) Let R be the region bounded by the graphs of y = sin(πx) and y = x3 - 4x, as shown

in the figure.

a) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is an isosceles triangle. Find the volume of this

solid.

b) The region R models the surface of a small pond. At all points in R at a

distance x from the y-axis, the depth of the water is given by h(x) = 3 - x.

Find the volume of water in the pond.

4) Region T is bounded by and . Cross-sections perpendicular to the y-axis are

rectangles whose height is four times its length in base T. Find the volume of the solid.

5) Region S is bounded by the ellipse. Cross sections perpendicular to the x-axis are isosceles

right triangles with hypotenuse in the xy-plane. Find the volume of the solid.