Chapter 8.3, Problem 70E

a.

To determine

To find : the area of the cross section.

Answer to Problem 70E

The area of the cross section that corresponds to the circumference is $2.3,1.6,1.5,2.1,3.2,4.8,7.0$ .

Explanation of Solution

Given information : The height of the vase is $6$

inches and the circumference of vase is shown below:

  

Calculation :

Using the circumference is given by $2\pi r$ .

Solve for $r$ in each cross section, and then use the formula for the area of a circle given by $\pi r^2$ .

  $2.3,1.6,1.5,2.1,3.2,4.8,7.0$ .

b.

To determine

To express : the volume of vase over the interval $\left[0,6\right]$ .

Answer to Problem 70E

The volume of vase as an integral with respect to y over the interval $\left[0,6\right]$ is $\frac{1}{4\pi }{\displaystyle \underset{0}{\overset{6}{\int }}C{\left(y\right)}^{2}dy}$ .

Explanation of Solution

Given information : The height of the vase is $6$

inches and the circumference of vase is shown below:

  

Calculation :

The volume of vase as an integral of the function $C\left(y\right)$ that means the circumference of the cross section at a given $y$ .

  $\frac{1}{4\pi }{\displaystyle \underset{0}{\overset{6}{\int }}C{\left(y\right)}^{2}dy}$

c.

To determine

To approximate : the integral by using trapezoidal rule.

Answer to Problem 70E

The value of integral by using trapezoidal rule is $34.7$ .

Explanation of Solution

Given information : The height of the vase is $6$ inches and the circumference of vase is shown below

  

Calculation :

Using the formula for the trapezoidal rule.

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx\approx \left(b-a\right)\frac{f\left(a\right)+f\left(b\right)}{2}}$

The area between consecutive cross section.

Adding these up gives you approximately $34.7$ square inches.