5.4 Integration formulas and the net change theorem  (Page 4/8)

${\displaystyle \int \left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx}$

${\displaystyle \int \left(e^{2x}-\frac{1}{2}{e}^{x\text{/}2}\right)dx}$

${\displaystyle \int \frac{dx}{2x}}$

${\displaystyle \int \frac{x-1}{x^2}dx}$

${\displaystyle {\int }_0^{\pi }\left(\text{sin}x-\text{cos}x\right)dx}$

${\displaystyle {\int }_0^{\pi \text{/}2}\left(x-\text{sin}x\right)dx}$

Write an integral that expresses the increase in the perimeter $P\left(s\right)$ of a square when its side length s increases from 2 units to 4 units and evaluate the integral.

Write an integral that quantifies the change in the area $A\left(s\right)=s^2$ of a square when the side length doubles from S units to 2 S units and evaluate the integral.

A regular N -gon (an N -sided polygon with sides that have equal length s , such as a pentagon or hexagon) has perimeter Ns . Write an integral that expresses the increase in perimeter of a regular N -gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

The area of a regular pentagon with side length $a>0$ is pa ² with $p=\frac{1}{4}\sqrt{5+\sqrt{5+2\sqrt{5}}}.$ The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area.

A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?

An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2 a units?

Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2 s units and evaluate the integral.

Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2 s units and evaluate the integral.