Reference no: EM13834334
Problem 1: Use the definition of an improper integral to evaluate the integral below:
![1442_1-definition of an improper integral.png](https://secure.expertsmind.com/CMSImages/1442_1-definition%20of%20an%20improper%20integral.png)
Problem 2: 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 f(x) = 1/ex is revolved about the x-axis
(b). The volume obtained when the area between the positive x-axis (x > 0) and the graph of f(x) = 1/ex is revolved about the y-axis.
Problem 3: Find the indefinite integral and evaluate the definite integral:
![542_3-indefinite integral.jpg](https://secure.expertsmind.com/CMSImages/542_3-indefinite%20integral.jpg)
Problem 4: Evaluate the integral as a sum of two integrals:
![2325_4-sum of two integrals.jpg](https://secure.expertsmind.com/CMSImages/2325_4-sum%20of%20two%20integrals.jpg)
Problem 5: Evaluate:
![932_5evaluate.jpg](https://secure.expertsmind.com/CMSImages/932_5evaluate.jpg)
Problem 6: (a) Suppose we divide the integral (1, 4) into 100 equally wide subintervals and calculate a Riemann sun for f(x) = 1 + x2 by randomly selecting a point ci 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 7: Sketch the graph of the integrand function and use it to help evaluate the integral
![2215_7-integrand function.jpg](https://secure.expertsmind.com/CMSImages/2215_7-integrand%20function.jpg)
Problem 8: (a) Find the area between f(x) = 1/x and the x-axis for 1 < x < 10, 1 < 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 9: 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.
![1237_9-Theorems of Pappus.jpg](https://secure.expertsmind.com/CMSImages/1237_9-Theorems%20of%20Pappus.jpg)
Problem 10: The figure below shows the graph of g. sketch the graph of g-1.
![1043_10-graph of g.jpg](https://secure.expertsmind.com/CMSImages/1043_10-graph%20of%20g.jpg)
Problem 11: Find the exact values of:
(a) tan(cos-1(x))
(b) cos(tan-1(x))
(c) sec(sin-2(x))
Problem 12: Use the definition of an improper integral to evaluate the given integral:
![375_12-definition of an improper integral.jpg](https://secure.expertsmind.com/CMSImages/375_12-definition%20of%20an%20improper%20integral.jpg)
Problem 13: Determine whether the improper integral is convergent of divergent. Do not evaluate the integral:
The volume obtained when the area between the x-axis for x > 1 and the graph of f(x) = sin(x)/x (in the figure below) is revolved about the x-axis,
![1727_13-improper integral.jpg](https://secure.expertsmind.com/CMSImages/1727_13-improper%20integral.jpg)
Problem 14: Complete the square in the denominator, make the appropriate substitution, and integrate:
![1673_14-square in the denominator.jpg](https://secure.expertsmind.com/CMSImages/1673_14-square%20in%20the%20denominator.jpg)
Problem 15: 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.
Problem 16: A car had the velocity given in the figure below. How far did the car travel from t=0 to t=30 seconds?
![2110_16-Car Velocity.jpg](https://secure.expertsmind.com/CMSImages/2110_16-Car%20Velocity.jpg)