Exam 6: Applications of Integration

Suppose a company has estimated that the marginal cost of manufacturing x items is $c ^ { \prime } ( x ) = 5 + 0.02 x$ (measured in dollars per unit) with a fixed start-up cost of c(0) = 10,000. Find the cost of producing the first 500 items.

Find the volume of the solid obtained by rotating the region bounded by the curves $y = \sqrt { 9 - x ^ { 2 } } , x = 0 , \text { and } y = 1$ about the x-axis.

Find the area of the region bounded by the curves $y = x ^ { 3 } - 2 x$ and $y = - x$

A curve is written parametrically as $x = 3 t - t ^ { 3 } , y = 3 t ^ { 2 }$ Find the arc length of the curve from $t = 0 \text { to } t = 1$ .

Let $f ( x ) = \ln x$ , find c such that $f _ { a v e } = f ( c )$ on the interval $[ 1 , e ]$ .

The base of a certain solid is an elliptical region with boundary curve $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ Cross-sections perpendicular to the x-axis are squares. Find the volume of the solid.

A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.

The tank in the figure below is full of water with a density 62.5 lb/ft³. How much work is required to empty the tank by pumping water to a point 4 feet above the top of the tank?

A force of 8 dynes is required to stretch a spring from its natural length of 10 cm to a length of 15 cm. How much work is done (a) in stretching the spring to a length of 25 cm? (b) in stretching the spring from a length of 20 cm to a length of 25 cm?

Find the volume of the solid obtained when the region bounded by the curves $y = x ^ { 2 } + 1 , y = 1 , \text { and } x = 1$ is rotated about the line x =1.

Find the volume of the solid obtained by rotating the region bounded by $y = \sqrt [ 3 ] { x } , y = e ^ { - 2 x }$ and x = 1 about the line x = 1. (Use a graphing calculator.)

A swimming pool 24 feet long and 15 feet wide has a bottom that is an inclined plane, the shallow end having a depth of 3 feet, and the deep end 10 feet. The pool is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pounds on the bottom of the pool.

Find the average value of the function whose graph is given below.

Find the volume of the solid obtained by rotating the region bounded by the curves $y = x ^ { 3 } , y = 0 , \text { and } x = 1$ about the line x = 2.

Let R be region bounded by the curve $4 y = x ^ { 2 } , x = 2 y - 4$ .(a) Find the volume of the solid obtained by rotating R about the x-axis.(b) Find the volume of the solid obtained by rotating R about the line x = 5.(c) Find the volume of the solid obtained by rotating R about the line y = -1.

A tank contains water. The end of the tank is vertical and has the shape below. Find the hydrostatic force against the end of the tank.

Set up, but do not evaluate, an integral for the length of $y = \tan x , 0 < x < \frac { \pi } { 4 }$

Find the area of the region bounded by the curves $y = ( x - 2 ) ^ { 2 } - 1$ and $y = 3 - x$ .

The marginal revenue for a company when sales are q units is given by $\frac { d R } { d q } = 12 - 0.06 q$ . Find the increase in revenue when the sales level increases from 100 to 200 units.

Find the volume of the solid obtained by rotating the region bounded by $y = \frac { 3 } { \sqrt { x ^ { 2 } + 9 } } , x = 4$ the y-axis and the x-axis about the y-axis.