Exam 6: Applications of the Definite Integral

When pumping water out of a full hemispherical basin with radius r feet, how far down will the water level be when 1/4 of the work has been done? [The density of water is 62.4 lbs/ft³.]

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

What is the maximum hydrostatic force a dam would need to withstand if it has the shape of a semicircle with a height (radius) of 50 feet? [The density of water is 62.4 lbs/ft³.]

An object is dropped from a height of 140 feet. Another object directly below the first is launched vertically from the ground with an initial velocity of 70 ft/s. Determine when and how high up the objects collide.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object , if the object thrown downward at a velocity of 6 feet per second from a height of 30 feet. [Take the origin to be on the ground.]

An underwater viewing window is rectangular with width 40 feet. The window extends from the surface of the water to a depth of 20 feet. Find the hydrostatic force on the window.

Let R be the region bounded by , the x-axis, and the y-axis. Compute the volume of the solid formed by revolving R about the y-axis.

A force of 7 pounds stretches a spring 7 inches. Find the work done in stretching this spring 10 inches beyond its natural length.

A cylindrical water tank has a radius of 4 feet and a height of 3.0 feet. Compute the work done to pump the water out of a filled tank through the top. [The density of water is 62.4 lbs/ft³.]

A projectile traveling with momentum 4 kg m/s in one direction undergoes an impulse in the opposite direction. The force of the impulse is described as 6000(sin500t) kg m/s² and it is exerted over a time range . What is the object's momentum (in the direction of the impulse) after the impulse?

Compute the volume of the solid formed by revolving the region bounded by and about the x-axis.

A 2-person tent has a rectangular base and rises to a single point 5' up. Compute the volume of the tent. 7 ft 8 ft

Identify the graph and area of the region bounded by the curves . The figures below are not necessarily to scale.

Identify the initial conditions y(0) and y'(0) for the vertical motion of an object, if the object is dropped from a height of 45 feet. [Take the origin to be on the ground.]

Where is the center of mass of a region of uniform density bounded by ?

Compute the arc length exactly. , -2 $\leq$ x $\leq$ -1

An object propelled from the ground with an initial velocity of 95 ft/s will reach a maximum height of 141.0 ft. If the initial velocity is increased 13%, by what percentage will the maximum height increase? Round percentages to the nearest integer.

Identify the region (solid lines) and axis of revolution (dashed line) for the solid whose volume is described by the integral .

Given a certain amount of land potentially devoted to farming, one will first use the more fertile regions where production costs are lowest before using the less fertile regions. The consequence of this logical strategy is that the production costs increase as production increases. A certain landowner has enough land to produce 10⁶ bushels of a particular crop, and the production costs, c, of the n^th bushel is described with the equation . If the market price is p for this particular crop, the profit for producing the n^th bushel is . If p = $5.25, how much total profit could the farmer expect to obtain by producing bushels.

The base of a solid V is the region bounded by and Find the volume if V has semicircular cross sections perpendicular to the x-axis.