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In this video I explain how to evaluate the indefinite integral Sin^2(Θ)dΘ, ∫ Sin²ΘdΘ. I demonstrate how to transform the integral using the trig identities Sin²Θ = 1/2(1-Cos2Θ), such that we have 2 separate functions which are simpler to integrate. I then demonstrate how to evaluate the integrals using U-Substitution and the power rule.

In this video I explain how to evaluate the indefinite integral Cos^2(Θ)dΘ, ∫ cos²ΘdΘ. I demonstrate how to transform the integral using the trig identities Cos²Θ = 1/2(Cos2Θ 1), such that we have 2 separate functions which are simpler to integrate. I then demonstrate how to evaluate the integrals using U-Substitution and the power rule

In this video I'd like to wish all my viewers a very merry Christmas and a Happy New Year for 2025

In this video I explain how to evaluate the indefinite integral Cos^4(Θ)dΘ, ∫ cos⁴ΘdΘ. I demonstrate how to transform the integral using the trig identities Cos²Θ = 1/2(Cos2Θ 1) and Cos²2Θ = 1/2(Cos4Θ 1), such that we have 3 separate functions which are simpler to integrate. I then demonstrate how to evaluate the integrals using U-Substitution

In this video I explain how to find the volume under a paraboloid and a disk using double integration and polar coordinates. I begin by defining the region of the disk and converting it from cartesian coordinates to polar coordinates. Once the region is defined I demonstrate how we can sum infinitesimally small pieces of area in the r direction to generate an infinitesimally small slice of area...

In this video I explain how to find the area of a segment in a circle. To understand how a segment is formed I begin by explaining what is meant by a sector of a circle and when a chord is drawn between the point where the sector intersects the circle, a segment is created. I also demonstrate that a triangle is created where the segment plus triangle make up the sector. Armed with this knowledg...

In this video I take a look at how to find the Volume between two surfaces using double integration and polar coordinates. I first look at defining the region of intersection between the two surfaces. I then demonstrate how to convert the region from cartesian (x, y) coordinates to polar coordinates (r, Θ). I then look in detail at the process of integrating in the r direction (the inner integr...

In this video I take a look at how to find the Volume under the surface of a Cone using double integration and polar coordinates. I first look at defining the region on top of which the cone is defined. I then demonstrate how to convert the region from cartesian (x, y) coordinates to polar coordinates (r, Θ). I then look in detail at the process of integrating in the r direction (the inner inte...

In this video I demonstrate how to find the arc length and sector area of circles. I first define what is meant by a circle sector and explain what is meant by a major and minor sectors and how both are portions or fractions of the total area of a circle. I then look at how to find the area of a minor circle sector given the subtended angle Θ in degrees and the radius of a circle. A brief demon...

In this video I take a look at how to find the area of a segment in a circle using double integration over a polar region. In the video I explain how to convert the equation of the circle to polar coordinates and how to convert the line that cuts it, creating the segment, into polar coordinates. So I convert the (x ,y) coordinates to (r, Θ) polar coordinates. I then take a detailed look at the ...

In this video we look at how to find the area of the region where two circles intersect using double integration The video explains how to convert the equation of each circles from cartesian coordinates (x, y) to polar coordinates (r, Θ). The video describes how to define the region to be integrated. It describes the angles which Θ need to rotate through in order to cover the total area of inte...

This video explains how to find the area of a region bounded by two circles using double integration. In this video we are given the equation of two circles in cartesian coordinates. We learn how to convert the equations to a system of polar coordinates, convert (x,y) to the polar system (r,Θ). The video then starts by defining the limits of integration in the r direction and provides a detaile...

This video evaluates the double integrals, utilizing polar coordinates, found in the video. th-cam.com/video/whbt-SnV0Dw/w-d-xo.html It then proceeds to use the results to find the center of mass of a circle with a missing segment. The inner integrals are evaluated first along r summing all the infinitesimally small slices dr, followed by the evaluation of the outer integral from 0 to alpha and...

This video explains how to find the region of a circle with a missing segment using double integration. This is part 1 of 2 and demonstrates how to formulate the double integrals. It explains how the region must be split into two parts, the first where the line cuts the circle between an angle of theta = 0 and theta = alpha and then from theta = alpha and theta = 90 degrees. It shows that this ...

Introduction to Long Division of Polynomials

Introduction to Homogeneous Differential Equations

Differential Equations | Liquid in Cylinder Problem | Separation of Variables

How to Solve Differential Equations By Separation of Variables | Particular Solution

How to Solve Differential Equations by Separation of Variables | General Solution

Identifying Linear and Nonlinear Ordinary Differential Equations

Verifying Solutions to Differential Equations

Introduction to Differential Equations

Maclaurin Series

Taylor Series

Fundamental Theorem of Calculus Part 2 | Proof | FTC2

Simpson's Rule

The Trapezium Rule | The Trapezoidal Rule

Mastering Integration by Partial Fractions | Calculus

The Mean Value Theorem for Integrals | The average Value of a function | Calculus