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Definite Integrals Review ( sections 5.6-5.10)

This quiz review covers topics including the Fundamental Theorem of Calculus, definite integral properties, area using integrals, and volumes of solids.

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Definite Integrals Review ( sections 5.6-5.10)

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  1. Definite Integrals Review(sections 5.6-5.10) AP Calculus

  2. Quiz Topics • Fundamental Theorem of Calculus (section 5.6) • Nderiv and FnInt on calculator • Definite integral properties (section 5.7) • Area using integrals (section 5.8) Example: Region bounded by y = -x^2 + 5 and y = -2x + 2 • Volumes of solids (section 5.9)

  3. Fundamental Theorem of Calculus To evaluate the definite integral of f(x) = from -1 to 2: Let F(x) = = Then = F(2) – F(-1): = = 3

  4. Integral Properties • If function values of f(x) are positive and the interval boundaries are increasing, the integral will be positive. • If functional values of f(x) are negative (below x axis) and the interval boundaries are increasing, the integral will be negative. • As a result, it is possible to have areas of positive and negative area “cancel,” resulting in an integral of 0.

  5. Odd/Even Integrals with Symmetric Limits Odd Function: f(-x) = -f(x) Even: f(-x) = f(x) ODD: EVEN: Examples: y = sin x, y = tan x Examples: y = cos x, ** Interval bounds must be symmetric!!!

  6. Reversal of limits of integration: dx Integral of Constant Times Function: dx Integral of Sum:

  7. Sum of Integrals With Same Integrand (Also allows integral to be broken into more convenient parts)

  8. Use Common Sense! dx

  9. Use Common Sense: Not all Integrals Involve Calculus! dx

  10. Area Bounded By Curves • Draw rectangular “strip” between curves and write formula for area using dx or dy for the width depending on the orientation. • May need to find intersection points of functions to find interval boundaries. Area of “strip” = l x w = (y1 – y2)dx Write in terms of x: Integral is the SUM of strips over interval:

  11. Area bounded by curves: “Sideways” • Functions on left and right instead of top and bottom. Make sure “strip” always extends from one function to another (Not one function back to itself)

  12. Integral Vs. Area • Be able to write the INTEGRAL EXPRESSION for the area.

  13. Additional Topics • Make sure you can find basic integrals (look over trig derivatives list – pay attention to SIGNS!) • Chain Rule for integrals: • “Tricky” integrals – change form, make graphs, etc: • Look out for odd/even functions (can graph to check) • Draw original function given f’(x)

  14. Additional Topics • Know how to find functional integral (fnInt) on graphing calculator • Be able to write equations for simple volume of solids (disc method) • Cylinder Volume: • Use trapezoidal rule/Riemann sums to estimate area using a table of values (no function given). (PLOT GRAPH!!!!) • Application problems: Area under curve for velocity-time graph is displacement! • Review Problems: pg. 261: R6 bcd, R7, R8, R10 (trap rule on part c)

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