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

Find the value of the solid that is obtained when the region under the curve y = x⁴ over the interval [2, 4] is revolved about the x-axis.

Approximate the arc length of y = sin 3x from x = 0 to x = $\pi$ .

Find the area of the surface generated when y² = 12x from x = 0 to x = 1 is rotated around the x-axis. Approximate area to three decimal places.

A trapezoidal gate in a dam that is 2 feet wide at the top, 4 feet wide at the bottom, and 6 feet tall is submerged in water (weight density 62.4 lbs/ft³) as shown in the figure. Find the force on the gate when the surface of the water is 2 feet above the top of the gate.

Answer true or false. It takes twice as much work to elevate an object to 210 m above the earth as it does to elevate the same object to 105 m above the earth.

Find the arc length of the curve y = 2x^3/2 from x = 0 to x = 4 .

Find the area of the region enclosed by y = 6x - x² and y = 5.

Answer true or false. The arc length of the curve y = x³ from x = 0 to x = 3 is 27.658.

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x², y = 0, and x = 3 is revolved about x = 3.

Answer true or false. It takes twice as much work to stretch a spring 64 cm as it does to stretch the same spring 32 cm.

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by y = 4x, x = 0, and y = 8 is revolved about x = 2.

Use cylindrical shells to find the volume of the solid that results when the first quadrant region enclosed by y = x⁵ and y = x is revolved about the y-axis.

Find the force (in N) on the top of a submerged object if its surface is 6.6 m² and the pressure acting on it is 4.6 * 10⁴ Pa. Neglect the effect of the atmosphere above the liquid.

An object in deep space is initially considered to be stationary. If a force of 620 N acts on the object over a distance of 575 m, how much work is done on the object?

Find the area of the region enclosed by x² = 4y and x - 2y + 12 = 0. Approximate area to three decimal places.

A spring exerts a force of 32 N when stretched 10 cm. How much work, in Joules, is required to stretch the spring from a length of 370 cm from its natural length to a length of 450 cm from its natural length?

Answer true or false. If when |x| > 0, .

Use cylindrical shells to find the volume of the solid when the region enclosed by , x = 2, x = 4, y = 0 is revolved about the y-axis.

Answer true or false. If a submerged rectangle is rotated 125° about an axis through its center and perpendicular to its surface, the force exerted on one side of it will be the same, provided the entire rectangle remains submerged.

Find the area of the surface generated when from y = 0 to y = 3 is rotated around the y-axis. Approximate area to three decimal places.