Exam 5: Applications of Integration

In a steam engine the pressure and volume of steam satisfy the equation , where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of and a volume of and expands to a volume of Use the fact that the work done by the gas when the volume expands from to volume is .

Find the indefinite integral.

Find the volume of a pyramid with height and base an equilateral triangle with side a = .

The base of S is a circular region with boundary curve . Cross-sections perpendicular to the x axis are isosceles right triangles with hypotenuse in the base. Find the volume of S.

Find the integral.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

BiFind the number b such that the line divides the region bounded by the curves and into two regions with equal area.

Use cylindrical shells to find the volume of the solid. A sphere of radius .

Find the volume common to two spheres, each with radius r = if the center of each sphere lies on the surface of the other sphere.

The base of a solid is a circular disk with radius . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

Find the area of the region bounded by the graph of f (x) = , the y-axis, and the tangent line to the graph of f at (16, 4).

Find the area of the region bounded by the given curves.

Evaluate the integral

Find the derivative of the function

In a certain city the temperature hours after 7 A.M. was modeled by the function Find the average temperature to three decimal places during the period from 7 A.M. to 7 P.M.

Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. The choices are rounded to the nearest hundredth.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis. y = x², y = 0, x = 3, x = 5; the x-axis

Find the derivative of the function.

Find the integral.

Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves. Rounded to the nearest hundredth.