What would go in the middle blank? **Title: Calculating the Surface Area of a Revolved Curve****Objective:**Learn how to write and evaluate the definite integral representing the area of the surface created by revolving a curve around the x-axis.**Given Function:**\[ y = \sqrt{81 - x^2} \]**Interval:**\[ -8 \leq x \leq 8 \]**Objective:**Calculate the surface area using the formula for surface area of revolution around the x-axis.**Surface Area Integral:**\[ 2\pi \int_{-8}^{8} \sqrt{81 - x^2} \, dx = 288\pi \]**Explanation:**- The function $ y = \sqrt{81 - x^2} $ can represent a semicircle with radius 9, due to the form of the equation.- The integral is evaluated from $-8$ to $8$, which is a segment of the semicircle.- Multiplying the integral by $2\pi$ accounts for the revolution around the x-axis, calculating the surface area of the resulting shape.- The evaluated integral yields a final result of $288\pi$.This approach provides the mathematical foundation for calculating the surface area formed when a curve is revolved around an axis, showcasing applications in geometry and calculus.

To find the value that the middle blank would fill.

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y = V 81 – x², -8 < x < 8 dx = 2887 J-8

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y = V 81 – x², -8 < x < 8 dx = 2887 J-8

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: Bella is reviewing a 1 inch to 2 foot scaled floor plan of her new house. The dimensions of her…

A: Given scale is 1 inch is 2 feet dimension of bedroom in scaled plane is 6 inches by 5 inches actual…

Q: Find all real solutions of the equation (x – 6)² = 25. with x1 < x2 and x2 Enter an integer or…

Q: Dalila keeps track of his favorite baseball players batting average, and notices the average is…

A: Given, Batting average=0.4727272 Attempts=55

Q: The cost of running the H & B Lawn Service is $350 a week plus $15.80 for each lawn mowed. If the…

A: Each lawn costs $15.80 So 32 lawns = (15.80*32)=$505.60

Q: Henry looks at the rectangle on the right and says that the base is 5cm and the height is 3cm.…

A: To write that who is correct

Q: A person has a rectangular board 10 inches by 16 inches around which she wants to put a uniform…

A: Given A person has a rectangular board 10 inches by 16 inches around which she wants to put a…

Q: windscreen by planting trees in a quarter-mile row. His daughter points out that 220 fewer trees…

Q: A rectangle has a width (W) that is ten centimeters less than three-eighths its length (L) and a…

Q: What will it cost to tile a rectangular floor measuring 295 feet by 65 feet if the tile cost $16 per…

Q: A police car is located 30 feet to the side of a straight road.

Q: 3. Find two positive numbers whose sum exceeds the sum of their cubes by the greatest amount.

Q: close his garden using 32 ers of fence. at is the largest interger value area, in are meteres, that…

Q: You have a box that measures 2 meters x 2 meters x 6 meters. What is the longest piece of rebar…

A: Given, you have a box that measures 2 meters ×2 meters×6 meters. we find the longest piece of rebar…

Q: Dan is ordering flats and would like to be able to fit at least 50 flats on his bench. Will he be…

A: 1foot=12inch 15foot=15*12=180inch 4.5foot=4.5*12=54inch Area of the bench = 180*54 = 9720 inch…

Q: If the taxi rate is 70 cents for the first mile and 40 cents for each additional mile,what is the…

Q: if i have 23 one foot square tiles how many packages do i need to cover a square ballroom floor…

A: It is given that one foot square tile contains 23 packages We have to find number of packages…

Q: Aira is celebrating her 20th birthday in style. She has taken into her Instagram Stories a series of…

A: Given query is to find the largest volume possible.

Q: What size would you see on the box for a new television whose screen measures 40 inches wide by 15…

Q: We have a rectangular piece of cardboard that is 27 centimeters by 37 centimeters. We form a box by…

Q: A worker’s wage depends on the number of hours he works. If his wage for 5 hours is ₱300, how much…

A: Wage

Question

100%

What would go in the middle blank?

**Title: Calculating the Surface Area of a Revolved Curve**

**Objective:**
Learn how to write and evaluate the definite integral representing the area of the surface created by revolving a curve around the x-axis.

**Given Function:**
\[ y = \sqrt{81 - x^2} \]

**Interval:**
\[ -8 \leq x \leq 8 \]

**Objective:**
Calculate the surface area using the formula for surface area of revolution around the x-axis.

**Surface Area Integral:**
\[ 2\pi \int_{-8}^{8} \sqrt{81 - x^2} \, dx = 288\pi \]

**Explanation:**
- The function $ y = \sqrt{81 - x^2} $ can represent a semicircle with radius 9, due to the form of the equation.
- The integral is evaluated from $-8$ to $8$, which is a segment of the semicircle.
- Multiplying the integral by $2\pi$ accounts for the revolution around the x-axis, calculating the surface area of the resulting shape.
- The evaluated integral yields a final result of $288\pi$.

This approach provides the mathematical foundation for calculating the surface area formed when a curve is revolved around an axis, showcasing applications in geometry and calculus.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

So the wind would be 350 or 450?
Suppose you have just enough money to buy a loaf of bread priced at $2.80. You have 16 coins, all quarters and dimes. How many quarters and dimes were there?
Vlad (the vampire) wanted to line the inside of his coffin bed with purple cloth. He measured the inside height, length and width to be twenty-three inches, eighty-four inches and twenty- eight inches. He also noted that it was a rectangular prism. The purple cloth is sold by the foot in sixty inch wide pieces. How many feet of cloth does he need to buy?
Q2 Mr Chong paid RM48 for 3 dictionaries and 4 pens whereas Mr Lee paid RM63 for 4 dictionaries and 5 pens. Find the price of a dictionary and of a pen
A doctor receives two invoices for an order including gloves and masks. The first order cost $605 and included 2000 gloves and 500 masks while the second order of 10000 gloves and 600 masks cost $916 dollars. The invoice does not show the cost of the gloves or masks. Find the cost of each glove and of each mask.
Luke is buying a computer monitor that is 12 inches tall and has a distance of 37 inches between opposite corners. What is the width of Luke's new monitor?
A field that is 3/4 acres must be planted with seed. If 24 people will split the job equally, how many acres must each person plant?Write your answer in simplest form.
How does the area of each rectangle relate to the length and the width? Why?
A photo is printed on a 20-inch by 24-inch piece of paper. The photo covers 320 square inches and has a uniform border. What is the width of the border?
Daniel received a citation and wants to frame the plaque in such a way as it creates a border of uniform width around the rectangular plaque. The picture company charges $1 per square inch for the size of the entire frame . If the plaque is 6 inches by 9 inches and Bonnie paid $70 to have it framed, how wide would the border be?