WebAssign Lesson 2-2 Volumes (Homework)

Transcription

1 WebAssign Lesson 2-2 Volumes (Homework) Current Score : / 38 Due : Thursday, July :00 AM MDT Jaimos Skriletz Math 175, section 31, Summer Instructor: Jaimos Skriletz 1. /2 points Find the cross sectional area and volume of the following objects 1. A prism where the cross-section is an equilateral triangle with side length of 2 in and length of 5 in. (Be accurate to 3 decimal digits.) Cross- sectional area: Volume: 2. A cylinder with radius of 11 cm and height 23 cm. (Be accurate to 1 decimal digit.) Cross- sectional area: Volume:

2 2. /2 points Find the cross sectional area and volume formulas for the following objects 1. A prism where the cross- section is a triangle as shown: Cross- sectional area: Volume: 2. A half-cylinder with radius r and height h. Cross- sectional area: Volume:

3 3. /3 points A pyramid has a 5 cm by 5 cm square base is 10 cm tall. Define a coordinate system at the tip of the pyramid as shown. 1. Find the infinitesimal amount of volume of a cross- sectional segment distance y from the tip of the pyramid as shown. dv = 2. Set up the definite integral and evaluate it to find the total volume. (Be accurate to at least one decimal place.) V = = dv

4 4. /3 points Find the volume created by rotating the curve y = x 2/3 about the x-axis between x = 0 and x = 8 were x and y are measured in inches. 1. Find the little bit of volume of a disc a distance x from the y-axis in terms of x. dv = 2. Set up the definite integral and find the total volume. (Be accurate to at least two decimal digits.) V = = dv

5 5. /3 points Find the volume created by rotating the curve = 1 where the linear distances are measured in feet. y = ln(x+1) about the y-axis between y = 0 and y 1. Find the little bit of volume of a disc a distance y from the x-axis in terms of y. dv = 2. Set up the definite integral and find the total volume. (Be accurate to at least two decimal digits.) V = = dv

6 6. /3 points Find the volume created by rotating the region bounded by where the linear distances are measured in meters. y = and y = 2 x about the x- axis 1. Find the little bit of volume of a washer a distance x from the y-axis in terms of x. dv = 2. Set up the definite integral and find the total volume. (Be accurate to at least 3 decimal digits.) V = = dv

7 7. /3 points Find the volume created by rotating the area bounded by where the linear distances are measured in millimeters. y = x 2 and y = x about the y-axis 1. Find the little bit of volume of a washer a distance y from the x-axis in terms of y. dv = 2. Set up the definite integral and find the total volume. (Be accurate to at least three decimal digits.) V = = dv

8 8. /2 points A cone has a height of h = 12 inches and a base of radius r = 8 inches. Let y be the distance from the base of the cone as shown. Answer the following questions in terms of this variable. 1. Set up the definite integral to find the volume of this cone. b V = a Where the bounds of integration are a = b = 2. What is the volume of this cone? (Be accurate to at least one decimal place.) 9. /2 points The area of an ellipse is πab, where a and b are the lengths of the semimajor and semiminor axes. Compute the volume of a cone of height h = 12 cm whose base is an ellipse with semimajor axis a = 6 cm and semiminor axis b = 4 cm. (Be accurate to at least two decimal digits.)

9 10. /2 points Calculate the volume of the solid obtained by revolving the region under the graph of f(x) = x + 8 about the x- axis over the interval [5, 6]. Linear distances are measured in centimeters. Be accurate to at least two decimal digits. 11. /2 points Calculate the volume of the solid obtained by revolving the region under the graph of f(x) = x 1 about the x- axis over the interval [4, 5]. Linear distances are measured in inches. Be accurate to at least 4 decimal digits.) 12. /2 points Find the volume of the solid obtained by rotating the region enclosed by the curves f(x) = x 2 and g(x) = x + 6 about the x-axis. Linear distances are measured in feet. Be accurate to at least two decimal digits.

10 13. /2 points Rotating the region in the figure below about the y-axis produces a solid with two different regions. Find the volume of the solid that is formed using centimeters for the linear distances. Be accurate to at least two decimal digits.

11 14. /2 points Find the volume of the solid obtained by rotating the region B in the figure below about the line y = 2. Assume linear distances are measured in inches. Be accurate to two decimal digits.

12 15. /2 points Find the volume of the solid obtained by rotating the region A in the figure below about the line x = 7. Assume linear distances are measured in inches. Be accurate to two decimal digits.

13 16. /3 points Find the volume of the cap to a sphere of radius R and height h as shown 1. Find the little bit of volume of a cross-sectional segment distance y from the center of the sphere. dv = 2. Set up the definite integral to find the total volume of this spherical cap. y=b V = dv y=a Where the bounds of integration are a = b = 3. What is the formula for the volume of this sphere cap? (WebAssign will accept non simplified answers.)