Find the area of the shaded region. Consider the given function and the given in

Question

Find the area of the shaded region. Consider the given function and the given interval. f(x) = (x - 7)^2, [6, 9] Find the average value f_ave of f on the given interval. Find c such that f_ave = f(c). Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y=ln 3x, y = 2, y = 3, x=0; about the y-axis Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 9x^6, y = 9x, x greaterthanorequalto 0; about the x-axis

Explanation / Answer

1)area of the shaded region =[-1 to 1]ey-(y2-5) dy

area of the shaded region =[-1 to 1]ey-(1/3)y3-5y +C

area of the shaded region =[e1-(1/3)13-5*1 +C]-[e-1-(1/3)(-1)3-5(-1) +C]

area of the shaded region =e-(1/3)-5 +C-e-1-(1/3)-5 -C

area of the shaded region =e -e-1-(2/3)-10

2)fave=[1/(9-6)][6 to 9](x-7)2 dx

fave=(1/3)[6 to 9](x-7)2 dx

fave=(1/3)[6 to 9](1/3)(x-7)3 +C

fave=[6 to 9](1/9)(x-7)3 +C

fave=[(1/9)(9-7)3 +C]-[(1/9)(6-7)3 +C]

fave=(1/9)(2)3 -(1/9)(-1)3

fave=(8/9) +(1/9)

fave=(9/9)

fave=1

b)fave=f(c)

1=(c-7)2

c-7=-1,c-7=1

c=6, smaller value

c=8 larger value

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