Exam 6: Applications of the Definite Integral

Two divers, one on the 15 ft platform and one on the 30 ft platform, hope to perform a joint dive in which they enter the water at the same time. If each begins with an upward jump with an initial velocity of at 14 ft/s, how much time difference should there be between the start of the two dives?

Find the mean of x, given the pdf , .

The pdf describing the chance that a given insect will die at a particular age, x days, is . What is the chance the insect will live longer than 6 days?

The moment of inertia, I, of an object is the second moment of its mass distribution relative to an axis of rotation, , where r is the perpendicular distance from the rotation axis and is the density of the object at a distance r from the rotation axis. What is the moment of inertia for rotation about the y-axis of a shape described by the region bordered by x = 0, and y = 0 revolved about the y-axis? Assume the density of the material is 1.0, so that mass and volume will be numerically equivalent. [Hint: Cylindrical shells are particularly conducive to moment-of-inertia calculations since r is everywhere the same for a given shell.]

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

A water tower is spherical in shape with radius 35 feet, extending from 120 to 190 feet above ground. Compute the work done in filling the tank from the ground.

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

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

Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by about x = 5.

Find the volume of the solid formed by revolving the region bounded by about y = 6.

A cylindrical water tank has a radius of 2 feet and a height of 6.0 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.]

Find the length of the curve defined by function and extending from x = 0 to x = 2. Round your answer to three decimal places.

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 mean of x, given the pdf .

Find the area of the region bounded by the given curves. Write a single integral that represents the area.

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

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.

Compute the arc length exactly.

Compute the arc length exactly.

Sketch the region determined by the intersections of the curves. Estimate the intersection points to the nearest integer, then use these points to estimate the area.