Lab F-5: Integration Calculator A calculator is either required or permitted on all of the following questions. When indicated, all written work must be shown to earn full credit. 1. Suppose that fis an integrable and even function such that the following are true. (7 points) If ( x ) dx = 2 if ( )dx = 1 I f ( * ) dx = 7 a ) Sf ( x ) dx 2' b ) S [ x 2 - 3f ( x ) ] dx c ) jf ( * ) dx2b 2 . Evaluate S |2x|dx in terms of b, where b is any real, positive number. (3 points) - b N f ( xx) 2 3 "' (x) 4 5 2 1/ 2 -1/2 3 3. The table above gives selected values for f, f' g, and g', where fand g are differentiable and integrable functions. (6 points) 2x a) Let / be the function given by k(x) = S g(t) dt and b be the function h(x) = 2x + sinx. For what values of x is h(x) = k'(1)? 9 (x ) b) Let m(x) = J f'(t) dt. Find the equation of the line tangent to m(x) where x = 2.4. To estimate the surface area of a pond, a surveyor takes several measurements, in feet, at 20-feet intervals, as shown in the figure. Estimate the surface area of the pond using trapezoids. (3 points) 501 541 821 821 73, 75; 80 5. A particle moves along the x-axis so that its velocity at any time t 2 0 is given by the function v(t) =- 1 + tcos(2nt). (7 points) a) Determine whether the particle is speeding up or slowing down at time t = 3/4. Justify your answer . b) Find the position of the particle at any time / if it is at x = 4 when t = 0. Show all work for full credit.6. Let fbe a differentiable function, defined for all real numbers x, with the following properties. f'(x) = ax + bx where a and b are real numbers f'(1) = 6 and f"(1) = 18 2 If ( x ) dx = 18 Find f(x). Show your work. (7 points) 7. The function b is given by h(x) = S (- 2 - 2in(Int) + (Int)')dt for 1. 1 s x