MATH 126 Calculus II Learning Goal Activity 5 7 and 6 1 Name Due Date September 1 2023 11 59pm On Canvas LEARNING GOAL to identify an appropriate substitution by rst recognizing that the integrand contains EITHER a function times a multiple of its derivative OR more generally a composition multiplied by a multiple of its inner function s derivative 1 Write a substitution equation and corresponding di erential one could use to evaluate each of the following b x2 2 x3 6x 7 8 dx integrals cid 90 arctan x a x2 1 dx cid 90 cid 90 cid 90 cid 90 cid 90 c y4 2 y5 7 dy t cid 112 ln t 7 dt d e sec2 tan2 d f cos u sin4 u du 1 LEARNING GOAL to evaluate an inde nite integral using the Change of Variables Formula 2 Use your lecture notes to help you evaluate the following integrals cid 90 cos a d sin3 cid 90 b 15x2 x3 1 3 2 dx cid 90 5 6t c 1 t2 dt LEARNING GOAL to use the Change of Variables Formula for De nite Integrals to rewrite an integral strictly in terms of a new variable via an appropriate substitution and corresponding change in the limits of integration 3 The graph of the arctangent function is depicted below Shade in a region whose area is equivalent to cid 90 e2 e arctan ln x x dx Be sure to justify your answers 4 3 2 1 1 2 3 4 x y 2 2 2 4 The Change of Variables Formula for De nite Integrals is used on cos x esin x dx to produce a new de nite integral Which of the following regions directly corresponds to that integral Select all such regions and be sure to justify your answer s cid 90 2 0 y C y A 1 y eu 2 u 1 y y sin u 1 B u y eu y D y 1 y 1 y sin u y F 4 2 u u 1 1 E 1 e 2 y u u u 2 2e LEARNING GOAL for De nite Integrals to compute a de nite integral via an appropriate substitution to use either the Fundamental Theorem of Calculus or the Change of Variables Formula 5 Use your lecture notes to help you evaluate the following integrals cid 90 2 2 4 a sin 2 x x dx cid 90 1 1 b t42t5 2 dt 3 MATH 126 Calculus II Learning Goal Activity 6 1 LEARNING GOAL to understand the various region descriptions horizontally simple and vertically simple 6 Label each of the following regions as either horizontally simple vertically simple both or neither y y y x x x Region description Region description Region description LEARNING GOAL to compute areas between two curves using integration techniques 7 Find the area of the region de ned by the curves y 8 x2 and y 3x2 8 4 8 Find the area of the region de ned by the curves and x 2 2y2 and x y2 1 9 Find the area of the region de ned by the curves y x2 2x and y x over the interval 0 4 Hint Draw a graph and check vertical vs horizontal simplicity May need to use more than one region 5