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Bài tập CALCULUS 46
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Created by T. Madas
Question 167
(***+)
y
f ( x ) = x2 − 2 x + 2
O
1
4
x
The curve C has equation
f ( x ) = x2 − 2 x + 2 , x ∈ » .
a) Find the area of the finite region bounded by C , the x axis and the straight
lines with equations x = 1 and x = 4 , shown shaded in the figure above.
b) Hence evaluate
4
2 f ( 5 − x ) dx .
1
MP1-O , 12 , 24
Created by T. Madas
Created by T. Madas
Question 168
(****)
y
y = 3x − 6
B
O
R
P
x
y = 4 x − x2
A
The figure above shows the graph of the curve C with equation
y = 4 x − x2 , x ∈ » ,
intersected by the straight line L with equation
y = 3x − 6 , 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 .
6
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 2 x m and is attached to one
side of the rectangle also measuring 2 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 2 , is given by
A = 60 x − 2 x 2 − 1 π x 2 .
2
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
