Exam 8: Techniques of Integration

The lifespan (in years) of a certain mammal is a random variable with exponential density. What is the probability that the mammal will live more than 6 years if the average lifespan is 4 years?

Evaluate the integral .

Verify that is a density function on and calculate its mean value.

Compute the integral ..

Use to calculate the approximation to the volume of the solid obtained by rotating the graph of over about the y-axis.

Evaluate .

To show that converges, we should use:

The partial fraction decomposition of is:

Use the substitution and the trigonometric identity to evaluate the integral.

Evaluate the integral .

Evaluate the integral .

Evaluate the integral .

Calculate and for the following. A) B)

Use the substitution and the identities and to rewrite the following integrand as a rational function and evaluate it.

Evaluate the integral .

A cereal-packaging company fills boxes on average with 40 oz of cereal. Due to machine error, the actual volume is normally distributed with a standard deviation of 0.3 oz. Let be the probability of a box having more than 41 oz. Express as an integral of an appropriate density function and compute its value numerically.

Evaluate the integral .

To evaluate the integral using Integration by Parts, we should choose:

Compute the integrals using the reduction formulas, or the substitution , and the identities , , as required. A) B)

Evaluate the following improper integrals, if they exist. A) B)