Exam 6: Applications of the Definite Integral in Geometry, Science and Engineering

Find the volume of the solid generated when the region between the graphs and g(x) = 0 over the interval [0, $\pi$ ] is revolved about the x-axis.

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves y = x³, x = 0, x = 3, y = 0 about the x-axis. Approximate area to three decimal places.

Find the centroid of the region bound by y = x + 3 and y = x² - 3.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁴, x = 0, and y = 3 is revolved about the y axis.

A cylindrical tank 6 feet in diameter and 6 feet high is filled with oil whose density is 48 lbs/ft³. How much work is required to pump the oil over the top of the tank?

Find the work done when a variable force of N in the positive x-direction moves an object from x = 4 m to x = 8 m.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by x = 8y-y² and x = 0 is revolved about the x-axis.

A 0.9-kg object is moving at 9.7 m/s. If a force in the direction of the motion does 15.8 J of work on the object, what is the object's final speed?

Find the centroid of the region bounded by the graphs of xy = 6 and x + y = 5.

Find the arc length of the curve x = y² - 6 over the interval y = 3 to y = 5. Approximate area to three decimal places. Use a CAS if necessary.

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = x², y = 0, and x = 1 is revolved about x = 1.

Find the arc length of the curve x = 2y² + 8 over the interval y = 5 to y = 7. Approximate area to three decimal places. Use a CAS if necessary.

A right triangle is submerged vertically with one side at the surface in a liquid of density $\rho$ . The triangle has a leg that is 5 m long located at the surface and a leg 5 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

Find the area between the curves x = |y|, x = -y + 8, y = 0. Approximate area to three decimal places.

Find if .

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 3x, x = 0, and y = 3 is revolved about x = 1.

Answer true or false. The force exerted by water on a surface of a square, vertical plate (suspended vertically) with edges of 4 ft if it is suspended with its top 4 ft below the surface is 5,990.4 lb. (The density of water is 62.4 lb/ft³.)

A spring whose natural length is 35 cm is stretched to a length of 40 cm by a 3-N force. Find the work done in stretching the spring.

A spring whose natural length is 20 cm is stretched to a length of 35 cm by a 2-N force. Find the work done in stretching the spring.

Use cylindrical shells to find the volume of the solid when the region enclosed by , x = 1, x = 4, y = 0 is revolved about the y-axis.