CALCULUS II [Type the document MATH 2414 subtitle] Turnell 2018 Review of Cal I Trig Derivatives: d d d [sinu ] [cosu ] [tanu ] dx dx dx d d d [secu ] [cotu ] [cscu ] dx dx dx dduddu1 ux() Exponential and Natural Log Functions: []eeuu and [lnu ] or dx dx dx u dx u 2 f ()xe x x f ()xx ln(sin()) d Power Rule: []xnn nx 1 dx f ()xx432 3 x 6 x x d Product Rule: []uv uv vu dx f ()xx 3 cos x du vuuv dT BTTB Quotient Rule: OR dx v v2 dx B B2 x3 1 fx() x2 2 Page 1 d n Chain Rule: unuu n1 dx 4 fx() 3 x2 5 x 1 f ()xx sin(5)3 fx() tan52 3 x 1 Page 2 Implicit Differentiation: dy Find , given xxyy3245439 dx dy Find , given cos(xy ) y3 4 dx Integrals: f ()xdx= a set of antiderivatives Why a set? Review of Basic Forms u n1 kdu ku C udun C n 1 n 1 sin(udu ) ___________________ csc(uudu )cot( ) ___________________ cos(udu ) ___________________ sec(uudu )tan( ) ___________________ sec2 (udu ) ___________________ csc2 (udu ) ___________________ u 1 edu ___________________ du ___________________ u 3 3 1 23 x 3 xdx5 x(5xdx 4) x Page 3 2 6x cos x 35dx 3 dx (4x 9) sin x tanx sec2 xdx xsin(6xdx2 ) x x2 dx 2xedx x2 1 Page 4 Calculus I Review Worksheet dy Find the derivative ()yor for each function. dx 2 2/3 cos(3x ) 1. y (x 4x 5) 6. fx() 2. y tan(2 x) x sin(3x ) 3. yx cos(5 ) 7. x3 2x2y2 5y3 10 4. y sin2 (3x) 8. x2 y 3tany 4 3 5. y x2 tan x 9. y cos(sin(x )) 10. y (2x 3)4 (3x 2)5 Find the indefinite or definite integral: sin x 11. dx 1. x(4xdx24 3) 1cos x dx 543 1 12. 2. x 2xx dx 4 x3 x 3 3. sec(5)tan(5)x xdx ee42xx1 13. dx dx ex 4. 72x 14. sin6 x cos xdx sin x 5. dx 2 cos3 x 15. tan(2x )sec (2xdx ) 6. sin 6xdx 16. 412x xdx2 dx 4 7. 17. sec (x )sec(xxdx ) tan( ) 13 x 3 2 8. xcos(xdx ) ex 18. dx 1x2 x 9. xedx e 1 1 x3 10. dx 3 1 x4 Page 5 Calculus I Review Worksheet-Answers 2(2x 4) 22 1. dy43 xy x y 1 7. 3(xx2 4 5) 3 dx 15yxy22 4 2 dy2 xy 2. yx 2sec(2) 8. dx x22 3sec y 3. yx 5sin(5 ) 9. yxx 323 cos( )sin(sin( x 3 )) 4. yxx 6sin(3 )cos(3 ) yx 15(3 2)44 (2 x 3) 8(2 xx 3) 35 (3 2) 5. yx 22sec ( x ) 2 x tan( x ) 10. yx (3 2)43 (2 x 3) (54 x 29) 3sin(3x )(xx sin(3 )) cos(3 x )(1 3cos(3 x )) 6. y (xx sin(3 ))2 (4x25 3) 2 1 3xx x 1. C 7. 13x C 13. eeeC 40 3 3 1 2 7 8. sin (x ) 4 sin(x ) C 14. C xxx65233 1 2 7 2. 1 2 2 C 1x 1 65 42x 9. eC 15. (tan2 (2x )) C 2 4 2 3 4 3 3 1 10. (1 x ) C 2 2 2 3. sec(5x ) C 8 16. 12 x C 5 3 11. ln 1 cos x C sec5 (x ) 1 1 17. C 4. ln 7x 2 C 12. C 5 7 3 3(x 3) 3 1 3 e 1 5. C 18. lnee 1 ln 1 ln 2cos2 x e 1 1 6. cos(6x ) C 6 Page 6 Area of Region Between Two Curves Review from Cal 1: The area under a curve. Write the integral or integrals to find the area of each region: 1. 2. 3. Area Between 2 Curves: 4. Page 7 5. With respect to x 6. With respect to y 7. 1 Page 8 Examples: Sketch a graph of the region bounded by the given equations and find the area of the region: y xx2 31 1. yx 1 y 2. fy() ,gy () 0, y 3 16 y2 (Use Desmos.com for graph) Larson 11th ed: p. 450: 7,15,17,19,23,24,35,37 Page 9 Page 10 Finding Distances Example 1: yx4 2 . Find the distance from the y-axis and x-axis in terms of x and y. Example 2: x yy2 4 . Example 3: x2 Find the distance from the y-axis in terms of x and y. Find the distance from y 3 to the line: 2 y 1 in terms of x and y. x2 Example 4: Find the distance from y 3 Example 5: Find the distance from yx 2 2 2 to the line: y 1 in terms of x and y. to the line: x 2 terms of x and y. Page 11 Example 6: Find the distance from yxx2,2 0 Example 7: Find the distance from yxx2,2 0 to the line: x 3 terms of x and y. to the line: y 8 terms of x and y. Example 7: Find the indicated distances: Example 8: Find the indicated distances: yx(2)3 2 xy(2),,6,1 2 yxyy Page 12 Volume: The Disk Method Review from Geometry: The volume of a cylinder 2 Vrh Determine the Volume of a Solid of Revolution: So, for the purposes of the derivation of the formula, let’s look at rotating the continuous function yfx () in the interval [,]ab about the x-axis. Below is a sketch of a function and the solid of revolution we get by rotating the function about the x-axis. Short animation: https://youtu.be/i4L5XoUBD_Q The volume of the disk (a cylinder) is given by: Vrx 2 . Approximating the volume of the solid by n disks of width x with radius rx(), produces: n 2 Volume of solid [(rxi )] x i1 This approximation appears to become better and better as x 0(n ). Therefore, n b 22 Volume of solid lim [rx (i )] x [ rx ( )] dx x 0 i1 a Page 13 As seen in animation, we can rotate functions around the y-axis: The radius is now a function of y. d Volume of solid [()]ry2 dy c Note: the radius is ALWAYS perpendicular to axis of rotation. Example 1: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis. yx4 2 Go to http://www.shodor.org/interactivate/activities/FunctionRevolution/ to see revolution. Example 2: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis. yx4 2 Page 14 Example 3: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis. x yy2 4 (p.461: 12) Revolving about a line that is NOT the x or y axis. Example 4: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region x2 about the line y 1: y 3 2 Example 5: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the line x 2: yx 2 2 (p.461: 14d) Page 15 Page 16 Volume: The Washer Method The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a representative washer. Volume of the washer ()R22rw Where R the outer radius and r inner radius If rotated around a horizontal axis, then b Volume of solid ([()]R xrxdx22 [()]) a If rotated around a vertical axis, then d Volume of solid ([()][()])R yrydy22 c Example 1: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by the graphs: yx 2 , yx , about x-axis. 3 (answer is ) 10 Page 17 Example 2: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by the graphs: yx 2 2 , y 0 , x 2 about the line y 8. (p.461: 14d) Example 3: Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by the graphs: yx 2 2 , y 0 , x 2 about the y-axis. (p.461: 14c) For the following problems, (a) Find the outer radius, R, and the inner radius, r, (b) find the limits of integration, and (c) set up the integral that gives the volume of the solid bounded by the given functions. For problems 1-3, also find the volume. Homework 1. yx 2 , yx , about the y-axis. 2. yx 2 , yx , about line x 1. 3. yx 2 , yx , about line y 3. 4. yx(1)12 , y 1, x 0 , x 1 about the x-axis. 5. yx(1)12 , y 1, x 0 , x 1 about line x 1. 6. yx(1)12 , y 1, x 0 , x 1 about line x 2. 7.2 p. 461: 5-15 odd, 19, 21, 29, 31 Page 18 Volume: The Shell Method Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by the graphs: yxx4 2 , y 0 , about y-axis.