6.1 Integration by Parts. 5 Examples

0:00 Introduction.

5:09 Example 1: Integration of x sin x

7:13 Example 2: Integration of ln x

8:39 Example 3: Integration of t^2 times e^t

12:50 Example 4: Integration of e^x times Sin x

12:59 Example 5: Integration of tan^-1 (x)

Find the Volume using a Washer.

Volume Using a Washer Three Examples in one.

How much work is required to pump all the water from a circular swimming pool over the side?

A circular swimming pool has a diameter of 24 ft, The sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all the water out over the side? (Use the fact that water weighs 62.5lb/ft^3).

Determine whether the Geometric Series is Convergent or Divergent. If it is conv., find the Sum.

Use the Comparison Test to determine whether the Series is Convergent or Divergent.

Sketch the Curve by using the Parametric Equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.

Determine whether the Series is Convergent or Divergent.

Find a power Series Representation for the Function. And Determine the Interval of Convergence.

Evaluate the indefinite integral as an infinite series.

Differentiate y = ln ( cosh (3x)).

Implicit Differentiation. Find dy/ dx for 2x^3+x^2y-xy^3=2.

Find the limit as x approaches " - infinity". L'Hospital's Rule " infinity * zero".

The Mean Value Theorem.

Let f(x) = -3x^2+2x+4. Find the value(s) of x that satisfy the Mean Value Theorem on the interval [ -1, 1].

Graph the Secant Line, the Tangent Line, and f(x).

Optimization. Find two positive numbers whose Product is 100 and whose Sum is a Minimum.

6.3 Partial Fractions. 5 Examples

I have Five Examples covering the Four Cases.

0:00 Introduction

2:52 Example 1: Integrate ( x^3+x)/(x-1) dx

5:16 Example 2: Integrate ( x^2+2x-1)/(2x^3+3x^2-2x) dx

16:43 Example 3: Integrate ( x^4-2x^2+4x+1)/(x^3-x^2-x+1) dx

26:00 Example 4: Integrate ( 2x^2-x+4)/(x^3+4x) dx

30.08 Example 5: Integrate ( 1-x+2x^2-x^3)/x(x^2+1)^2 dx

Find the Volume using Cylindrical Shells.

Find the Volume Using Cylindrical Shells.

Find the VOLUME of the solid generated by rotating the region bounded by the given curves about the specified axis.

Sketch the (2-D) region and a typical Disk, Washer, or Cylindrical Shell. Y= x^4, y = 0, x = 1, about x = 2.

Determine whether the Sequence Converges or Diverges. If it converges, find the limit. an=(n+2)!/n!

Determine whether the Geometric Series is Convergent or Divergent. If it is conv., find the Sum.

Test the Series for Convergence or Divergence

Sketch the Curve by using the Parametric Equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. Eliminate the parameter to find a Cartesian Equation of the Curve.

Test the Series for Convergence or Divergence.

Test the Series for Convergence or Divergence.

Differentiate the following y = e^(cosx) + cos(e^x)

Differentiate f(x) = x^2 ln(2x+1).

Differentiate y=SQR ( 1+4 sin(x)). And find equations of the tangent line and the normal at (0, 1).

Find the linearization of the function f(x) = SQR(x+5) at a = 4 and use it to approximate SQR (3.8).

6.5 Approximate Integration_5 Examples.

Approximate Integration_5 Examples.

Riemann Sums.

Left End Point Approximation.

Right End Point Approximation.

Midpoint Approximation.

Trapezoid Rule.

ERROR BOUNDS.

Simpson's Rule.

Arc Length Example. Y= ln (cos x) from 0 to pi/3

First Order Linear Differential Equation Example.

Solve the initial value problem.

Y’ +2x^-1 y = 4x, y (1) = 2.

Determine whether the Sequence Converges or Diverges. If it converges, find the limit. {n^2 e^(-n)}.

Use the Integral Test to Determine whether the Series is Convergent or Divergent.

Test the Series for Convergence or Divergence.

Determine whether the series is convergent or divergent.

Determine whether the series is Absolutely Convergent, Conditionally Convergent, or Divergent.

Find the Maclaurin Series of f(x) and find the associated Radius of Convergence.

Differentiate y = 3^ (x ln x)

Differentiate y = x ^(SQR x).

Find the limit as x Approaches 0 of (e^4x -1-4x)/x^2.

L'Hospital's Rule_ Intermediate form (0/0).