See the attached file for questioins

Pblem 1:

(a) Suppose we divide the interval [1, 4] into 100 equally wide subintervals and calculate a Riemann sum for f(x) = 1 + x² by randomly selecting a point c_i in each subinterval. We can be certain that the value of the Riemann sum is within what distance of the exact value of the area between the graph of f and the interval [1, 4]?

(b) What if we take 200 equally long subintervals?

Problem 2:

Sketch the graph of the integrand function and use it to help evaluate the integral

            The graph of the integrand function looks like this:

Problem 3:

(a) Find the area between f(x) = 1/x and the x-axis for 1 ≤ x ≤ 10, 1 ≤ x ≤ 100, and

1 ≤ x ≤ A. What is the limit of the area for 1 ≤ x ≤ A as A→∞?

(b) Find the volume swept out when the region in part (a) is revolved about the x-axis for 1 ≤ x ≤ 10, 1 ≤ x ≤ 100, and 1 ≤ x ≤ A. What is the limit of the volumes for 1 ≤ x ≤ A as A → ∞ ?

Problem 4:

The center of a circle in the figure below with radius r is at the point (0, R). Use the Theorems of Pappus to find the volume and surface area swept out when the circular region is rotated about the x-axis.

Pblem 5:

The figure below shows the graph of g. Sketch the graph of g−1.

Problem 6:

Find the exact values of:

(a) tan(cos^-1(x))

(b) cos(tan^-1(x))

(c) sec(sin^-1(x))

Problem 7:

Use the definition of an improper integral to evaluate the given integral:

Problem 8:

Determine whether the improper integral is convergent or divergent. Do not evaluate the integral:

Problem 9:

Complete the square in the denominator, make the appropriate substitution, and inte- grate:

Problem 10:

Use the tube method to calculate the volume when the region between the x-axis and

the graph of y = sin(x) for 0 ≤ x ≤ π is rotated about the y-axis. ***** Please go to next page for the bonus question *****

*** EXTRA CREDIT QUESTION ***

Problem 11: A car had the velocity given in the figure below. How far did the car travel from t = 0 to t = 30 seconds?

Problem 12:

Use the definition of an improper integral to evaluate the integral below:

Problem 13:

Determine whether the improper integral is convergent or divergent. Do not evaluate the

integral.

(a) The volume obtained when the area between the positive x-axis (x ≥ 0) and the

graph of  is revolved about the x-axis.

(b) The volume obtained when the area between the positive x-axis (x ≥ 0) and the graph

of is revolved about the y-axis. (Use the method of “tubes” from section 5.5).

Problem 14:

Find the indefinite integral and evaluate the definite integral:

Problem 15:

Evaluate the integral as a sum of two integrals:

Problem 16:

Evaluate: