Exploring Applications of Integrals (Calculus Guide)

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Area Between Two Curves

1 hr 23 min 7 Examples

Overview of finding area between two curves
Finding area between curves given the limits of integration and when limits must be determined (Example #1 and #2)
Dividing the region and using the y-axis to find area between curves (Example #3 and #4)
Introduction to how area between curves relates to finding volume

1 hr 31 min 9 Examples

Volumes of known cross sections: formulas and overview
Set up (do not evaluate) the integral for volume using a known cross section (Examples #1-2)
Set up the integral to find the volume using cross sections (Examples #3-4)
Find the volume whose cross sections are rectangular heights (Examples #5a-b)
Determine the volume of x²+y²=4 whose cross sections are squares (Example #6)
Find the volume whose cross sections perpendicular to the y-axis are semicircles (Example #7)
Determine the volume whose cross sections are isosceles right triangles (Example #8)
The base of the solid is a triangular region with given vertices; find the volume using a known cross section (Example #9)

1 hr 52 min 10 Examples

Solids of Revolution using the Disk Method
Find the volume by revolving bounded region about x-axis (Example #1)
Determine the volume formed by revolving bounded region about x-axis (Examples #2-3)
Calculate the volume formed by revolving bounded region about y-axis (Examples #4-5)
Find the volume by revolving bounded region about y=k (Examples #6-7)
Find the volume by revolving bounded region about x=h (Examples #8-9)
Use two integral statements to find the volume by revolving about x-axis (Example #10)

2 hr 26 min 7 Examples

Solids of Revolution – Washer Method Formulas
Find the volume of the solid about the axes (Example #1a-b)
Use the washer method to find the volume about the axes (Example #2a-b)
Find the volume of the solid formed by revolving about the y-axis (Example #3)
Use the washer method to find the volume about y=k (Example #4a-b)
Use the washer method to find the volume about x=h (Example #4c-d)
Calculate the volume of the solid about the x-axis (Example #5a)
Calculate the volume of the solid about the line x=-1 (Example #5b)
Set up the integral to find the volume of the solid about axis of revolution (Example #5c-e)
Find the volume if a circle is revolved about horizontal line (Example #6)
Use the disk and washer method to find the volume (Example #7)

1 hr 38 min 9 Examples

Cylindrical Shell Method Overview and Formulas
Comparing disk and shell method (Example #1 and #2)
Using shell method to find volume of solids about vertical and horizontal axes (Examples #3-6)
Finding volume using cylindrical shell method and volume of y=sinx about y-axis (Examples #7-9)

1 hr 26 min 7 Examples

Arc Length Formula for Calculus
Set-up the integral to find the length of the smooth curve (Example #1a-c)
Calculate the length of the curve (Examples #2-3)
Find the arc length over the given interval (Example #4)
Overview of Surface Area of Revolution Formulas
Find the resulting surface area generated by revolving about the x-axis (Example #5)
Find the surface area about the x-axis (Example #6)
Calculate the surface area about the y-axis (Example #7)

2 hr 5 min 15 Examples

Introduction to force, work, and constant force
Find the work done by a constant force (Examples #1-2)
Understanding work when the force or distance changes (Example #3)
How much work is done lifting a leaky bucket from the ground by a rope (Example #4)
How much work is done lifting a leaking sandbag (Example #5)
Find the work done by a rocket lifting off the ground (Example #6)
Work done pulling a chain to the top of a building (Example #7)
How much work is needed to pull a rope coiled on the ground through a window (Example #8)
Understanding work on a spring — Hooke’s law
How much work is done stretching a spring from its natural length (Example #9)
How much work is done stretching or compressing a spring (Examples #10-11)
How far beyond its natural length can the spring be stretched when a force is applied (Example #12)
Understanding work due to pumping
How much work is required to pump all the water out of a cylindrical pool (Example #13)
Find the work required to empty a hemisphere-shaped tank of water (Example #14a)
Find the work required to empty a hemisphere-shaped tank with a spout above the tank (Example #14b)
How much work is required to pump water out of a conical tank with a spout above the tank (Example #15)

1 hr 51 min 9 Examples

Introduction to Video: Hydrostatic Force
Overview of Hydrostatic Force exerted on an object
Find the fluid force of a rectangular plate submerged even to surface (Example #1)
Find the fluid force of a triangular plate submerged even to surface (Example #2)
Determine the fluid force of a trapezoidal plate submerged even to surface (Example #3)
Determine the fluid force of a semicircular plate submerged even to surface (Example #4)
Determine the fluid force of a parabolic plate submerged even to surface (Example #5)
Find hydrostatic force of a square plate submerged below the water’s surface (Example #6)
Find hydrostatic force of a right triangular plate submerged below the water’s surface (Example #7)
Determine the hydrostatic force of a square plate submerged below the surface (Example #8)
Find hydrostatic force against the semicircular gate of partially submerged dam (Example #9)

34 min 2 Examples

Overview of what is the Center of Mass (centroid) and Moments
Example #1 of finding the center of mass using integration by parts
Example #2 of finding the center of mass using Integration by Parts and Half-Angle Identity

2 hr 44 min 24 Examples