Exam 6: Applications of the Definite Integral

Molding is typically long, thin, decorative pieces of wood used, for instance, around windows or doors. The cross-section of a particular type of molding can be described as the region bounded by , x = 4, x = 0, and (all measured in inches), projected perpendicular to the xy-plane (see figure). If the manufacturer paints all four sides (but not the ends) and each gallon of paint covers 250 ft², how many linear feet of molding can be coated with one gallon?

When a certain spring is stretched 0.20 m, 7 of work is done. What is the spring's "spring constant"?

Find the area between the following curves on the given interval.

A ball is thrown at an angle of with an initial speed of 25 meters/second. What is its time of flight ? [Assume the ball is thrown from ground level and lands at ground level.]

A solid shape described as the region bounded by revolved around the y-axis, needs to have 1/4 of its volume trimmed off to meet a weight restriction. This will be done by reducing the radius of the piece - in other words, the resulting shape will be described as above, but with the region also bounded by x = q. Find the resulting radius, q.

A crash test is performed on a vehicle. The force of the wall on the front bumper is shown in the table. Estimate the impulse and the speed of the vehicle to the nearest whole number (use m = 200 slugs).

A water tower is spherical in shape with radius 50 feet, extending from 100 to 200 feet above ground. Compute the work done in filling the tank from the ground.

A certain not-so-wiley coyote discovers that he just stepped off the edge of a cliff. Fourteen seconds later he hits the ground in a puff of dust. How high in meters is the cliff?

What is the total hydrostatic force on the walls of a cylindrical water tower 50' high and having a radius of 18'? [The density of water is 62.4 lbs/ft³.]

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Set up the integral for arc length and then approximate the integral with a numerical method. Round answers to four decimal places.

Identify the integral used to determine the surface area of the surface of revolution for the shape described by , , revolved about the x-axis.

Find the area of the region determined by the intersections of the curves.

Estimate the intersection points to the nearest tenth then use these points to estimate the area.

Set up the integral for the surface area of the surface of revolution and approximate the integral with a numerical method. Round answers to two decimal places. revolved about the x-axis

A projectile passes overhead, and you attempt to shoot it down with a cannon. When it passes overhead it is traveling horizontally at 100 m/s at an elevation of 1000 m. Your cannon launches shells at 200 m/s, and in this instance you fire just as the projectile passes overhead and at an angle of 60^o. In this scenario the horizontal velocities of the two projectiles are the same. At what elevation will the cannon shell and the other projectile collide?

A 0.5 kg rock is dropped from the edge of a 52-foot cliff. How fast will it be going when it hits the ground at the base of the cliff?

Use the best method available to find the volume of the solid formed by revolving the region bounded by and about the x-axis. Write the integral that is used to find the volume.

Find the volume of the solid with cross-sectional area extending over the range .

A football (American) has a circular cross section with radius approximately for . Compute the volume of the football.