Exam 7: Techniques of Integration

Evaluate the integral using the indicated trigonometric substitution.

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places.

A body moves along a coordinate line in such a way that its velocity at any time t, where , is given by . Find its position function if it is initially located at the origin.

Find the integral.

Find the integral using an appropriate trigonometric substitution.

Evaluate the integral to six decimal places.

Determine whether the improper integral converges or diverges, and if it converges, find its value.

Use the Table of Integrals to evaluate the integral to three decimal places.

Let a and b be real numbers. What integral must appear in place of the question mark "?" to make the following statement true?

Find the integral.

Find the integral using an appropriate trigonometric substitution.

Find the integral.

Use the Trapezoidal Rule to approximate for . Round the result to four decimal places.

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with and the formula to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, is the concentration of the dye in the aorta, and R is measured in liters per minute. Round to one decimal place.

Find the integral.

Evaluate the integral.

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with subintervals.

Evaluate the integral or show that it is divergent.

Evaluate the integral.

Evaluate the integral.