Exam 5: Integration

Find the average value of the function $f ( x ) = \frac { x } { 4 x + 1 }$ over the interval [1, 3]. Round your answer to two decimal places.

Determine the area of the region bounded by the curve $y = x ^ { 3 } - x ^ { 2 } - 2 x + 4$ and the line y = 4.

Find the consumer's surplus for a commodity whose demand function is $D ( q ) = 30 e ^ { - 0.03 q }$ dollars per unit if the market price is $p _ { 0 } = \$ 21$ per unit. (Hint: Find the quantity q₀ that corresponds to the given price p₀ = D(q₀).)

Given an initial population P₀ = 8,000, a renewal rate $R ( t ) = 300 e ^ { 0.04 t }$ and a survival function $S ( t ) = e ^ { - 0.025 t }$ , where t is measured in years, determine the population at the end of 7 years.

Determine the area of the region bounded by the curve $y = x ^ { 3 } - x ^ { 2 } - 2 x + 2$ and the line y = 2.

The chief economist for a meat-packing plant estimates that if L worker-hours of labor are employed each week, the number of pounds of meat made ready for sale will be given by $Q ( L ) = 600 L ^ { 3 / 4 }$ . Find the level of labor that would result in the average production when labor varies from 500 to 700 hours. Round to the nearest whole hour.

Find the area of the region under the line y = 4x + 9 over the interval 1 $\le$ x $\le$ 9 .

Given an initial population, $P _ { 0 } = 110,000$ , a renewal rate, R = 197, and a survival function, $S ( t ) = e ^ { - 0.05 t }$ , with time t measured in years, determine the population at the end of 18 years.

Evaluate $\int \left( x ^ { 6 } - 7 x ^ { 2 } + 3 \right) d x$ .

Evaluate $\int \left( e ^ { - 3 x } + \frac { 5 } { x } \right) d x$ .

Evaluate $\int _ { 1 } ^ { 2 } \left( 1 - \frac { 3 } { x } + \frac { 5 } { x ^ { 3 } } \right) d x$ .

Evaluate $\int x ^ { 7 } e ^ { x ^ { 2 } + 3 } d x$ .

Evaluate $\int \left( 5 x ^ { 6 } - 9 x + 7 \right) d x$

It is estimated that t days from now a farmer's crop will be increasing at the rate of $0.3 t ^ { 2 } + 0.6 t + 1$ bushels per day. By how much will the value of the crop increase during the next 8 days if the market price remains fixed at $6 per bushel?

In a certain section of the country, the price of chicken is currently $3 per kilogram. It is estimated that x weeks from now, the price will be increasing at a rate of $3 \sqrt { x + 1 }$ cents per kilogram. How much will chicken cost 9 weeks from now?

Determine the area of the region bounded by the curves $y = 6 + 4 x - x ^ { 2 }$ and $y = 6 + x ^ { 2 }$ .

Evaluate $\int _ { 0 } ^ { 1 } \frac { x } { \left( 3 x ^ { 2 } - 7 \right) ^ { 2 } } d x$ .

Find the area of the region under the line y = 8x + 1 over the interval 1 $\le$ x $\le$ 4 .

The average value of $f ( x ) = x ^ { 2 } - 3 x + 5$ over the interval 1 $\le$ x $\le$ 3 is $\frac { 10 } { 3 }$ .