Bài tập CALCULUS 46

Created by T Madas Question 167 (***+) y f ( x ) = x2 − x + O x The curve C has equation f ( x ) = x2 − x + , x ∈ » a) Find the area of the finite region bounded by C , the x axis and the straight lines with equations x = and x = , shown shaded in the figure above b) Hence evaluate  f ( − x ) dx MP1-O , 12 , 24 Created by T Madas Created by T Madas Question 168 (****) y y = 3x − B O R P x y = x − x2 A The figure above shows the graph of the curve C with equation y = x − x2 , x ∈ » , intersected by the straight line L with equation y = 3x − , x ∈ » As shown in the above figure, C meets L at the points A and B , and crosses the x axis at the origin O and at the point P The finite region R is bounded by C , L and the x axis Show that the area of R , shown shaded in the figure, is 19 C2A , proof Created by T Madas Created by T Madas Question 169 (****) 2x y The figure above shows the design of a theatre stage which is the shape of a semicircle attached to rectangle The diameter of the semicircle is x m and is attached to one side of the rectangle also measuring x m The other side of the rectangle is y m The perimeter of the stage is 60 m a) Show that the total area of the stage, A m , is given by A = 60 x − x − π x b) Show further, by using a differentiation method, that the maximum area of the stage is 1800 m2 π +4 SYN-Y , proof Created by T Madas