A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 19, 59, 77, 94, 106, 118, 128, 64, 38, and 0. Use the Midpoint Rule with n=5n=5 to estimate the volume VV of the liver.

 

I need help solving some problems including area bounded by aregion and volume by cylindrical shells.

1) y = 3/x

y=12x

y=1/12x

x>0

2)
Find thevolume V of the solid obtained by rotating the regionbounded by the given curves about the specifiedline.

y=4-(1/2x)

y=0

x=2

x=3

rotated about xaxis

3)Find thevolume V of the solid obtained by rotating the regionbounded by the given curves about the specified line.
x = 4sqrt(5y)

x=0

y=3

rotated on y axis

4)Find the volume V of the solid obtained by rotatingthe region bounded by the given curves about the specifiedline.
y = ln3x

y =4

y =5

x = 0

rotated on yaxis

5)
Find the volume V of the solid obtained by rotating theregion bounded by the given curves about the specified line.

y =3x⁵

y =3x

x ≥ 0

rotated on X axis

6)
Find the volume V of the solid obtained by rotating theregion bounded by the given curves about the specifiedline.

A cross-section of an airplane wing is shown. Measurements ofthe wing, in centimeter, at 19-sentimeter intervals are 6.3, 20.6,26.9, 29.0, 27.2, 26.8, 23.8, 20.3, 16.2, 8.2, 3.1. Use themidpoint rule with n = 5 to estimate the area of the wing'scross-section if a = 190. (Assume the thickness of the edges if0).

______ cm^2.

(a) Suppose S is a solid with known cross-sectional areas. Explain how to approximate the volume of S by a Riemann sum. Then write an expression for the exact volume.

(b) If S is a solid of revolution, how do you find the cross-sectional areas?