C3
Worksheet C
INTEGRATION
1
y
y=
O
12
(2 x + 1)3
1
x
The diagram shows part of the curve with equation y =
12
(2 x + 1)3
.
Find the area of the shaded region bounded by the curve, the coordinate axes and the line x = 1.
2
y
y=9−
O
7
x
− 2x
x
The diagram shows the curve with equation y = 9 −
7
x
− 2x, x > 0.
a Find the coordinates of the points where the curve crosses the x-axis.
b Show that the area of the region bounded by the curve and the x-axis is 11 14 − 7 ln
3
7
2
.
a Sketch the curve y = ex − a where a is a constant and a > 1.
Show on your sketch the coordinates of any points of intersection with the coordinate axes
and the equation of any asymptotes.
b Find, in terms of a, the area of the finite region bounded by the curve y = ex − a and the
coordinate axes
c Given that the area of this region is 1 + a, show that a = e2.
4
y
P
y=e
x
O
Q
x
R
The diagram shows the curve with equation y = ex. The point P on the curve has x-coordinate 3,
and the tangent to the curve at P intersects the x-axis at Q and the y-axis at R.
a Find an equation of the tangent to the curve at P.
b Find the coordinates of the points Q and R.
The shaded region is bounded by the curve, the tangent to the curve at P and the y-axis.
c Find the exact area of the shaded region.
 Solomon Press
C3
INTEGRATION
Worksheet C continued
5
The function f is defined by
f(x) ≡ (
3
x
− 4)2, x ∈ , x > 0.
a Find the coordinates of the point where the curve y = f(x) meets the x-axis.
b Find the values of the constants p, q and r such that
f(x) = px−1 + qx
− 12
+ r.
The finite region R is bounded by the curve y = f(x), the line x = 1 and the x-axis.
c Show that the area of R is approximately 0.178
6
y
4
y=
2
5− x
1
O
x
The diagram shows the curve with equation y =
2
5− x
, x < 5.
a Show that the equation of the curve can be written as x = 5 −
2
y
.
The shaded region is bounded by the curve, the y-axis and the lines y = 1 and y = 4.
b Show that the area of the shaded region is 15 − ln 16.
7
y
4x + 5
y=
O
1
5
x
4x + 5 .
The diagram shows the curve with equation y =
Find the area of the shaded region bounded by the curve, the x-axis and the lines x = 1 and x = 5.
8
y
A
B
y = x(7 − 3x)
y=
4
x
O
x
The diagram shows the curves y = x(7 − 3x) and y =
4
x
, x > 0, which intersect at the
points A and B.
a Show that at A and B, 3x3 − 7x2 + 4 = 0.
b Find the coordinates of A and B.
c Find the area of the finite region enclosed by the two curves, giving your answer in the
form a + b ln 2, where a and b are rational.
 Solomon Press