**Problem Statement:**Find the surface area of the volume generated when the curve $ y = x^2 $ revolves around the y-axis from $ (1, 1) $ to $ (3, 9) $.**Graph/Diagram Description:**The image includes a visual representation of the volume generated, resembling a concave shape that opens wider as it extends upward, similar to a vase or a bowl. **Multiple-Choice Options:**1. $ \frac{\pi}{6} \left( 41\sqrt{41} - 5\sqrt{5} \right) $2. $ \frac{\pi}{6} \left( 37\sqrt{37} - 3\sqrt{3} \right) $3. $ \frac{\pi}{6} \left( 37\sqrt{37} - 5\sqrt{5} \right) $4. $ \frac{\pi}{6} \left( 37\sqrt{37} - 5\sqrt{5} \right) $The choices are given as options with circles in front, indicating that a user can select one of them.**Explanation:**To determine the surface area of the solid of revolution, one would generally use the method involving the integral that accounts for surface area. Even though just the multiple choices are provided and the problem is about choosing the correct one, normally you would calculate the surface area by integrating the function with respect to the axis of revolution (the y-axis in this case).The problem provides four different options based on this integration method. Given the complexity typically involved in solving such a problem, it is useful to confirm the computations through detailed calculus techniques involving parameterization and surface area formulas.

Answered: Image /qna-images/answer/21a0af2b-9974-4603-9eb8-fe510c75ed2c.jpg

Answered: Find the surface area of the volume…

Find the surface area of the volume generated when the curve y =x revolves around the y-axis from (1, 1) to (3, 9). ○종 (41V4T-5v5) ○중 (37v/37-3v5) Ox (37/37 – 5v5) ○등 (37v37-5v5)

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...

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Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Question

**Problem Statement:**

Find the surface area of the volume generated when the curve $ y = x^2 $ revolves around the y-axis from $ (1, 1) $ to $ (3, 9) $.

**Graph/Diagram Description:**

The image includes a visual representation of the volume generated, resembling a concave shape that opens wider as it extends upward, similar to a vase or a bowl.

**Multiple-Choice Options:**

1. $ \frac{\pi}{6} \left( 41\sqrt{41} - 5\sqrt{5} \right) $
2. $ \frac{\pi}{6} \left( 37\sqrt{37} - 3\sqrt{3} \right) $
3. $ \frac{\pi}{6} \left( 37\sqrt{37} - 5\sqrt{5} \right) $
4. $ \frac{\pi}{6} \left( 37\sqrt{37} - 5\sqrt{5} \right) $

The choices are given as options with circles in front, indicating that a user can select one of them.

**Explanation:**

To determine the surface area of the solid of revolution, one would generally use the method involving the integral that accounts for surface area. Even though just the multiple choices are provided and the problem is about choosing the correct one, normally you would calculate the surface area by integrating the function with respect to the axis of revolution (the y-axis in this case).

The problem provides four different options based on this integration method. Given the complexity typically involved in solving such a problem, it is useful to confirm the computations through detailed calculus techniques involving parameterization and surface area formulas.

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