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Form 4 Mathematics area and approximation: Integration questions and answers
Find the area bounded by the curve
y
=
2
x
3
-
5
, the x-axis and the lines x=2 and x=4.
(2m 4s)
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1.
(a) The gradient of the curve
y
=
a
x
2
+
b
x
at the origin is equal to 8. Find the values of a and b if the curve has a maximum point at x=4. (b) Determine the area of the region bounded by the lines x=0, x=6, y=0 and the curve
y
=
x
2
+
b
x
, for the values of a and b obtained in part (a).
2.
Determine the area bounded by the curve
y
=
x
2
-
4
, the x-axis and the line x=4.
3.
Find the area enclosed by the curve
y
=
4
x
-
x
2
, the x-axis and the lines x=1 and x=2.
4.
Find the area bounded by the curve
y
=
2
x
3
-
5
, the x-axis and the lines x=2 and x=4.
5.
(a) Find the values of x at which the curve
y
=
x
2
-
2
x
-
3
crosses the x-axis. (b) Find
∫
(
x
2
-
2
x
-
3
)
d
x
(c) Find the area bounded by the curve
y
=
x
2
-
2
x
-
3
, the x-axis and the lines x=2 and x=4.
6.
The curve of the equation
y
=
2
x
+
3
x
2
, has
x
=
-
2
3
and x=0 as x-intercepts. The area bounded by the curve, x-axis,
x
=
-
2
3
and x=2 is shown by the sketch below. Find: (a)
∫
(
2
x
+
3
x
2
)
d
x
. (b) The area bounded by the curve, x-axis,
x
=
-
2
3
and x=2.
7.
The gradient function of a curve is given by the expression 2x+1. If the curve passes through the point (-4, 6); (a) Find: (i) The equation of the curve (ii) The values of x at which the curve cuts the x-axis. (b) Determine the area enclosed by the curve and the x-axis.
8.
(a) Complete the table below for the function
y
=
x
2
-
3
x
+
6
in the range
2
≤
x
≤
8
(b) Use the trapezium rule with 10 strips to estimate the area bounded by the curve,
y
=
x
2
-
3
x
+
6
, the lines x=-2, x=8, and the x-axis. (c) Use the mid-ordinate rule with 5 strips to estimate the area bounded by the curve,
y
=
x
2
-
3
x
+
6
, the lines x=-2, x=8, and the x-axis. (d) By integration, determine the actual
9.
Evaluate
∫
4
2
x
2
+
2
x
-
15
d
x
10.
Find the value of k if
∫
2
0
(
k
x
3
–
3
x
2
)
d
x
=
16
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