Calculus II (MAT 146) Dr. Day Friday, February 2, 2018 Volumes of Solids with Known Cross Sections (6.2 & 6.3) Quiz #5 Today! Return to Methods of Integration (Ch 7) Integration Method #2: Integration by Parts (7.1) Test #1: Friday, Feb 9: STV 211 Friday, February 2, 2018 MAT 146

Friday, February 2, 2018 MAT 146

Friday, February 2, 2018 MAT 146

Friday, February 2, 2018 MAT 146

Volumes of Solids of Revolution (6.2 & 6.3) Dynamic Illustration #1 (discs) Dynamic Illustration #2 (washer) Dynamic Illustration #3 (shell) Dynamic Illustration #4 (cross section I) (cross section II) Friday, February 2, 2018 MAT 146

Region R in the first quadrant of the xy-plane is bordered by the x-axis, the line x = 4, and the curve y = √x. Determine the volume of the solid of revolution generated when R is rotated about the line y = 2. Determine the volume of the solid of revolution generated when R is rotated about the line x = −1. (A) (8pi)/3 (B) (544pi)/15 Friday, February 2, 2018 MAT 146

Friday, February 2, 2018 MAT 146

Consider the first-quadrant region R with borders y = sin(x) y = 0 and x = π/2 Sketch region R on the xy-plane. Calculate the exact area of R. Show evidence to support your solution. Set up, but do not calculate, a definite integral to represent the volume of the solid created when R is revolved around the y-axis. Set up, but do not calculate, a definite integral to represent the volume of the solid created when that solid is built on a base R with cross sections, perpendicular to the x-axis, that are semi-circles. Friday, February 2, 2018 MAT 146

Consider the first-quadrant region R with borders y = sin(x) y = 0 and x = π/2 Sketch region R on the xy-plane. Friday, February 2, 2018 MAT 146

Consider the first-quadrant region R with borders y = sin(x) y = 0 and x = π/2 Calculate the exact area of R. Show evidence to support your solution. Friday, February 2, 2018 MAT 146

Consider the first-quadrant region R with borders y = sin(x) y = 0 and x = π/2 Set up, but do not calculate, a definite integral to represent the volume of the solid created when R is revolved around the y-axis. Shells: Washers: Friday, February 2, 2018 MAT 146

Consider the first-quadrant region R with borders y = sin(x) y = 0 and x = π/2 Set up, but do not calculate, a definite integral to represent the volume of the solid created when that solid is built on a base R with cross sections, perpendicular to the x-axis, that are semi-circles. Friday, February 2, 2018 MAT 146

MAT 146

Undoing the Product Rule Friday, February 2, 2018 MAT 146

Integration by Parts Key Component of Integrand’s Two Factors For at least one factor, its derivative is “simpler” than the factor. For at least one factor, its anti-derivative is no more complex than the factor. Friday, February 2, 2018 MAT 146

Friday, February 2, 2018 MAT 146

You Choosing U: A Decision Algorithm L: log functions I: inverse trig functions A: algebraic functions T: trig functions E: exponential functions MAT 146 Friday, February 2, 2018

Review: Derivatives of Inverse Trig Functions Friday, February 2, 2018 MAT 146

Derivatives of Inverse Trig Functions: Co-Function Connections Friday, February 2, 2018 MAT 146

Trig Integrals Friday, February 2, 2018 MAT 146

More Trig Integrals! Friday, February 2, 2018 MAT 146

Which Roots Can You Simplify Mentally? Friday, February 2, 2018 MAT 146

Pythagorean Trig Identities Friday, February 2, 2018 MAT 146

Integrals Begging for Trig Substitutions! Friday, February 2, 2018 MAT 146

Using Trig Substitutions Friday, February 2, 2018 MAT 146

Trig Substitutions Friday, February 2, 2018 MAT 146

Integration Strategies: Which of These Can You Evaluate, WITHOUT Your CAS? Friday, February 2, 2018 MAT 146

Partial-Fraction Decomposition ?????? Friday, February 2, 2018 MAT 146

Partial-Fraction Decomposition Friday, February 2, 2018 MAT 146

What About… Improper Algebraic Fractions? Unfactorable Quadratics? Repeated Linear Factors? Friday, February 2, 2018 MAT 146