Mathematics & Statistics Texas Tech University Kent Pearce Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu

 Math 1452 Calculus II Spring 2015 Riddle Calculus Smith, Strauss, Toda

Review Exam I

 Section Content Suggested Problems Section 6.1 Area between two curves y = f(x) and y = g(x), a ≤ x ≤ b, f(x) ≥ g(x) on [a,b] Process Subdivide the area into vertical strips Approximate area of vertical strip by rectangular area, Rk Rk ≈ [f(xk) - g(xk)]Δ(xk) Form Riemann sum of area approximations using rectangles Limiting case as |Δ(xk)| → 0 Integral solution: Subcases needed for area bounded between intersecting curves Area using horizonatal strips Page 429: 13, 16, 18, 19, 25, 28, 30 Section 6.2 Volume of solids with known cross-sectional area Volume of solids of revolution (about x-axis or y-axis) Disk/Washer method Revolve area between two curves y = f(x) and y = g(x) about x-axis f(x) ≥ g(x) ≥ 0 on [a,b] Integral solution: Shell Method Revolve area between two curves y = f(x) and y = g(x) about y-axis f(x) ≥ g(x) ≥ 0 on [a,b] where a ≥ 0 Integral solution: Volumes of solids of revolution for areas bounded by two curves x = F(y) and x = G(y) Volumes of solids of revolution (about other horizontal or vertical lines) Page 443: 1, 3, 13, 15, 21, 26, 30, 34, 35 Section 6.3 Polar coordinate system Non-uniqueness of polar coordinates Converting polar coordinates to rectangular coordinates Converting rectangular coordinates to polar coordinates Graphs of equations given in polar coordinates Standard graphs Page 451 Intersection points of graphs of equations given in polar coordinates Polar area Area of a sector Integral solution: Page 456: 3, 4, 10, 12, 16, 23, 24, 31, 42, 44, 49 Section 6.4 Arc length of curve y = f(x) on [a,b] Integral solution: Arc length element Arc length of curve x = F(y) on [c,d] Area of surface of revolution of curve y = f(x) on [a,b] About the x-axis About the y-axis Integral solution: where R(x) is the distance from the curve to the axis of rotation and ds is the arc length element Area of surface of revolution (about other lines horizontal or vertical lines) Arc length of curve given by polar equation r = f(θ) Integral solution: Arc length element Area of surface of revolution of curve given by polar coordinates r = f(θ) (about x-axis or y-axis) Integral solution: where R(θ) is the distance from the curve to the axis of rotation and ds is the arc length element Page 466: 5, 8, 15, 23, 25, 30, 38

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