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C2 INTEGRATION Answers - Worksheet A page 2 Solomon Press 6 a = 2 ∫ 0 (x 2 + 2) dx b = 1 ∫ −2 (3x2 + 8x + 6) dx = [13 x3 + 2x]2 0 = [x 3 + 4x2 + 6x] 1 −2 = (83 + 4) − 0 = 2 3 6 = (1 + 4 + 6) − (−8 + 16 − 12) = 15 c = 4 ∫ 2 (9 + 2x − x 2) dx d = 0 ∫ −1 (x3 − 4x + 1) dx= [9x + x2 − 31 3 x]4 2 = [1 4 x4 − 2x2 + x] 0 −1 = (36 + 16 − 64 3
C2 INTEGRATION Answers - Worksheet A page 2 ... total area = 8 3 + 5 12 = 1 12 3 shaded area = 6 ...
Integration can be used to calculate areas. In simple cases, the area is given by a single definite integral. But sometimes the integral gives a negative answer which is minus the area, and in more complicated cases the correct answer can be obtained only by splitting the area into several parts and adding or subtracting the appropriate integrals.
Calculus: Integrals, Area, and Volume Notes, Examples, Formulas, and Practice Test (with solutions) Topics include definite integrals, area, “disc method”, volume of a solid from rotation, and more. Mathplane.com
CHAPTER 5 WORKSHEET INTEGRALS ANSWER KEY Trapezoidal Rule 1. a) Using a trapezoidal rule 0 2 4 16 36 64 176 2 4 8 8 0 0 2 ³ x dx | b) The answer from (a) is an overestimate because the graph of y x2 is concave up (notice that y" 2! 0.) Therefore the trapezoids yield “extra” area. c) 3 2 170 8 0 ³x2dx. The exact area is smaller than the ...
Use your expression in a and integration to find an estimate for the area of R. Use the trapezium rule with 6 equally-spaced ordinates to find another estimate for the area of R. 8= x2 + 16xR4 xThe diagram shows the curve. = x2 + 16 for x > 0.Show that the stationary point on the curve.
There are two ways to solve this problem: we can calculate the area between two functions and using the vertical elements and integrate with respect to x, or we can use the horizontal elements and calculate the area between the y -axis and the function integrating the functions with respect to y.
