Exam 13: Definite Integrals - Techniques

Use the function from to and n equal subintervals with the function evaluated at the right-hand endpoints of each subinterval. Let the sum of the areas of the rectangles be S. Find by using the formula for the sum of the areas of the n rectangles. ​

Use integration by parts to evaluate the integral ​

Use an integral formula to evaluate . ​

When the area under from to is approximated, the formulas for the sum of n rectangles using left-hand endpoints and right-hand endpoints are: ​ Left-hand endpoints: ​ Right-hand endpoints: ​ Find and . ​

Use the Trapezoidal Rule to approximate with n = 4. Round your answer to 3 decimal places. ​

Suppose that the Carter Car Service franchise finds that the income generated by its stores can be modeled by assuming that the income is a continuous stream with a monthly rate of flow at time t given by (dollars per month). Find the total income from a Carter Car Service store for years 3 through 5 of operation and round to the nearest penny. ​

Use an integral formula to evaluate . ​

Use the function from to and n equal subintervals with the function evaluated at the right-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S). ​

Approximate the area under the curve defined by the function over the interval x = 0 to x = 3 using the left-hand endpoints of three subintervals (rectangles). ​

Suppose in a small city the response time t (in minutes) of the fire company is given by the probability density function . For a fire chosen at random, find the probability that the response time is 10 minutes or less. Round your answer to three decimal places. ​

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to in pounds per year. Suppose also that the waste decays exponentially at a rate of 6% per year. Then the amount of radioactive waste that will accumulate in b years is given by and this integral evaluates to . How much waste will accumulate in the long run? Take the limit as in the integral evaluated. Round your answer to the nearest pound, if it exists. ​

The cost of producing x units of a certain item is Find the average value of the cost function C(x) over the interval from 0 to 1,000. Round answer to nearest penny. ​

Evaluate the definite integral . ​

In a manufacturing process involving several machines, the average down time t (in hours) for a machine that needs repair has the probability density function . Find the probability that a failed machine's down time is 4 hours or more. Round your answer to three decimal places. ​

Find the numerical value of by using the sum formulas. ​

Use rectangles to find the area between and the x-axis from to ​

The following table shows the rate of oil consumption (in thousands of barrels per year) by a certain city. Estimate the total consumption of oil by the city from 1999 -2004 by using 5 equal subdivisions and left-hand endpoints to estimate the area under the graph that corresponds to the table from 1999 to 2004.

Evaluate the definite integral . ​

Use integration by parts to evaluate the integral ​

If the demand function for a product is and the supply function is where p is in millions of dollars and x is the number of thousands of units. Find the consumer's surplus. Round your answer to the nearest million dollars. ​​