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Chapter 6: Q. 41 (page 511)

Consider the region between the graph of fx=4-x2and the liney=5on 0,2For each line of rotation given in Exercises 41鈥�44, use definite integrals to find the volume of the resulting solid.

Short Answer

Expert verified

The volume of the solid is15蟺

Step by step solution

01

Step 1. Given Information

The given figure is

fx=4-x2=y

x=4-y=gy

02

Step 2. Finding Volume

V1=蟺鈭�04(2-4-y2dyV1=蟺鈭�04(8-y-24-y)dy=蟺8y-y22+24-y323/204V1=11蟺

and

V2=蟺鈭�4522dy=4蟺鈭�45dyV2=4蟺5-4=4蟺

Thus, Final volume is

V=V1+V2V=11蟺+4蟺=15蟺

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Most popular questions from this chapter

Suppose an object is heating up according to a model for Newton鈥檚 Law of Cooling with temperature satisfying dTdt=k(350鈭�T)for some constant k.

(a) What is the ambient temperature of the environment under this model?

(b) Given that the temperature T(t) is increasing and that 0<T<350, is the constant kpositive or negative, and why?

(c) Use the differential equation to argue that the object鈥檚 temperature changes are faster when it is much cooler than the ambient temperature than when it is close to the ambient temperature.

(d) Part (c) is the basis for the oft-misunderstood saying 鈥淐oldwater boils faster.鈥� Why?

Write the volume of the two solids of revolution that follow in terms of definite integrals that represent accumulations of disks and/or washers. Do not compute the integrals.

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

dydx=3x+1xy

dydx=5y,y0=1

Consider the region between the graph of f(x)=3xand the x-axis on [1,3]. For each line of rotation given in Exercises 31鈥� 34, use definite integrals to find the volume of the resulting solid.
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