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Chapter 14: Q. 1TB (page 1118)

Area: Finding the area of a region in the x y-plane is one of the motivating applications of integration. It is also a special case of the surface area calculation developed in this section. Find the area of the region in the x y-plane bounded by the curvesy=x2andx=y.

Short Answer

Expert verified

a

Step by step solution

01

Step 1. Given Information.

Given two equations of curves:y=x2andx=y.

02

Step 2. Area calculation.

a

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Most popular questions from this chapter

Compute dS for your parametrization in Exercise 7.

Find the areas of the given surfaces in Exercises 21鈥�26.

S is the lower branch of the hyperboloid of two sheets z2=x2+y2+1that lies below the annulus determined by 1鈮�r鈮�2 in the xy plane.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing 鈭�Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then 鈭�Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then 鈭�SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing 鈭�SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing 鈭�SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.

Give a smooth parametrization, in terms of u and v, of the sphere of radius k and centered at the origin.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.F(x,y,z)=xeyzi+yexzj+zexyk
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