Is:f(x) dx

45. Let r(t) be the rate at which the world's oil is consumed,where t is measured in years starting at t = a on January I,2000, and r(t) is measured in barrels per year. What doesJ; r(t) dt represent?46. A radar gun was used to record the speed of a runner at thetimes given in the table. Use the Midpoint Rule to estimatethe distance the runner covered during those 5 seconds.t (5) V (m/s) r (5) v (m/s)0 0 3.0 10.510.5 4.67 3.5 10.671.0 7.34 4.0 10.761.5 8.86 4.5 to.812.0 9.73 5.0 10.812.5 10.22[lli]50. The Fresnel function Sex) = f~sin01Tt 2 ) dt was introducedin Section 5.3. Fresnel also used the functionin his theory of the diffraction of light waves.(a) On what intervals is C increasing?(b) On what intervals is C concave upward?(c) Use a graph to solve the following equation correct to twodecimal places:(d) Plot the graphs of C and S on the same screen. How arethese graphs related?47. A population of honeybees increased at a rate of r(t) bees perweek, where the graph of r is as shown. Use the MidpointRule with six subintervals to estimate the increase in the beepopulation during the first 24 weeks.X IX f(t)( fer) dt = x sin x + --, dr.0 0 1 + t-2 [Xf(t) dt = 2 sin x - IJ"2 J: f(x)f'(x) dx = [f(b)]2 - [f(a)]216 20 24 r(weeks)54. Find !im -.!.. f2+h JJ+t3 dt.h~O h 2-X-I if-3""'x""'0f(x) = { " .-.J]'=7If a "'"x "'" IEvaluate (3 f(x) dx by interpreting the integral as adifference of areas.fo'f(x) dx = fo'f(I - x) dx49. If fis continuous and J~f(x) dx = 6, evaluateJ~"'/2f(2 sin 6) cos 6 d6.