Volume

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AP Calculus AB/BC - 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

The volume of a sphere is given by V = (4/3)πr^3. If the radius is increasing at a rate of 4 cm/s, what is the rate of change of the volume with respect to time when the radius is 3 cm?

A bottle and a can are both being filled with water at a varying rate. The volume of water in the bottle is twice the volume of water in the can plus 20 milliliters. At time t=3 seconds, the can is being filled at a rate of 8 mL/s. How quickly is the bottle being filled at t=3 seconds?

A drop of water in space is currently in the shape of a cylinder with radius 1 mm and height 2 mm. A piston pushes the circular faces of the drop closer together, decreasing the height at a rate of 1 mm/s. If the water stays in the shape of a cylinder and its volume does not change, at what rate is the radius of the drop increasing?

The length of a rectangular prism is increasing at a rate of 6 in/s, the width is increasing at a rate of 4 in/s, and the height is decreasing at a rate of 2 in/s. If the length is 8 in, the width is 6 in, and the height is 4 in, what is the rate of change of the volume with respect to time?

A company wants to manufacture open-top rectangular boxes with a volume of 1000 cm³. What should be the dimensions of the box to minimize the amount of material used for its construction?

When a three-dimensional object can be broken down into an infinite series of two-dimensional shapes what technique can we use to find the volume of the solid?

Which formula can be used to find the volume of a solid with square cross sections over an interval [a,b]?

The graphs of y = x^2 -4 and y = 2x - x^2 create a bounded area that is the base of a solid. This solid has cross sections that are perpendicular to the 𝑥-axis and form squares. What are the bounds of the integral for the volume of this solid?

Which formula can be used to find the volume of a solid with rectangular cross sections over an interval [a,b]?

The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. What are the numerical bounds for the integral that can be used to find the volume of this solid when cross sections are taken perpendicular to the x-axis?

The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. What is the function A(x) that can be used to find the volume of this solid when cross-sections are taken perpendicular to the x-axis?

The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. Which integral can be used to find the volume of this solid when cross-sections are taken perpendicular to the x-axis?

The base of a solid with square cross sections is bounded by y = \sqrt{x-1}, x = 3, y = 0. What is the volume of this solid?

If the equation V = \int_0^2(y^3)^2 dy is the volume formula for a solid with square cross sections taken perpendicular to the y-axis, what is the function A(y) for the area of one square cross section?

If the equation V = \int_0^2(y^3)^2 dy is the volume formula for a solid with square cross sections taken perpendicular to the y-axis, what is the formula for the length of one side of the square?

What method is used to find the volume of a solid with triangular or semicircular cross sections?

Which integral gives the volume of a solid with equlateral triangle cross sections over an interval [a,b]?

Which integral gives the volume of a solid with semicircular cross sections over an interval [a,b]?

Consider a region defined by the function f(x) = x^3, revolved around the x-axis from x = 1 to x = 2. What is the volume of the solid formed by this revolution?

Consider a region defined by the function f(x) = x^2, revolved around the x-axis from x = 0 to x = 1. What is the volume of the solid formed by this revolution?