Take your time . kindly solve correctly in the order to get positive feedback.i promise you I will give you upvote if you solve all parts correctly. I can't repost it for separate parts because all parts are related with eachother Figure 2: Graph of the solid of revolution obtained by rotating the function f(x) = 4 - x²in the interval [0, 2] around the x-axis. Deep Dives1. 'Round and 'round we go (#integration)1-2/5Our goal in this part is to develop a framework for computing the volume of a specincclass of solids, specifically those that are symmetric and can be constructed as a solidof revolution. We begin by considering a function f(x) in the interval [a, b]. As ourfirst example, let's take the function f(x) = 4x², in the interval [0, 2], as shownbelow in Figure 1. The solid of revolution obtained by revolving f(x) around thex-axis is shown in Figure 2.(a) What shape is obtained if we take a vertical slice from the solid of revolution,say at x = 1? What is the area of such slice? How do the shape and its areachange if we look at a vertical slice through x = 0?(b) Find a general formula for the area of a vertical slice at any value x. Explain allthe terms in your formula.1(c) We can think of the volume of the solid of revolution as a "stack" of all theseslices. We should thus be able to compute its total volume by summing up thevolume of each of these slices. In order to do that, we must first assign somethickness to each slice¹, which we will denote Ar. Write an expression for thevolume of:(i) an individual slice at any value r(ii) the sum of all slices if we have n total slices.(d) What limit expression would give us the exact volume of the solid?(e) Find the integral expression that is equivalent to the limit expression you foundabove. Justify your reasoning and explain the values of the bounds on the inte-gral.(f) Use this expression to compute the volume of the solid of revolution shown inFigure 2.(g) Would the formula change if we had revolved around the y-axis instead? Do youexpect the volume of the solid of revolution obtained by revolving f(x) aroundthe y-axis to be larger, smaller, or the same as that from part (f)?(h) [The Method of Disks] Generalize: given any function g(x), find the formulaof the volume of the solid of revolution obtained by revolving g(x) around the (i)x-axis; (ii) y-axis.(You may assume g(x) has an inverse g-¹(y)).Figure 1: Graph of the function f(x)=4-2² in the interval [0, 2].⠀⠀

The given function is fx=4-x2 in the interval 0,2 which is rotated around the x-axis to generate a…

1-2/5 1. 'Round and 'round we go (#integration) Our goal in this part is to develop a framework for computing the volume of a specinc class of solids, specifically those that are symmetric and can be constructed as a solid of revolution. We begin by considering a function f(x) in the interval [a, b]. As our first example, let's take the function f(x) = 4-2², in the interval [0, 2], as shown below in Figure 1. The solid of revolution obtained by revolving f(x) around the x-axis is shown in Figure 2. (a) What shape is obtained if we take a vertical slice from the solid of revolution, say at x = 1? What is the area of such slice? How do the shape and its area change if we look at a vertical slice through x = 0? (b) Find a general formula for the area of a vertical slice at any value z. Explain all the terms in your formula.

1-2/5 1. 'Round and 'round we go (#integration) Our goal in this part is to develop a framework for computing the volume of a specinc class of solids, specifically those that are symmetric and can be constructed as a solid of revolution. We begin by considering a function f(x) in the interval [a, b]. As our first example, let's take the function f(x) = 4-2², in the interval [0, 2], as shown below in Figure 1. The solid of revolution obtained by revolving f(x) around the x-axis is shown in Figure 2. (a) What shape is obtained if we take a vertical slice from the solid of revolution, say at x = 1? What is the area of such slice? How do the shape and its area change if we look at a vertical slice through x = 0? (b) Find a general formula for the area of a vertical slice at any value z. Explain all the terms in your formula.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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