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Form 4 Mathematics area and approximation: Integration questions and answers
Evaluate #int _2^4 x^2 + 2x - 15\ dx#
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1.
(a) The gradient of the curve #y=ax^2 + bx# 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 + bx#, 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=4x-x^2#, the x-axis and the lines x=1 and x=2.
4.
Find the area bounded by the curve #y=2x^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^2-2x-3# crosses the x-axis. (b) Find #int(x^2-2x-3)dx# (c) Find the area bounded by the curve #y=x^2-2x-3#, the x-axis and the lines x=2 and x=4.
6.
The curve of the equation #y=2x + 3x^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) #int(2x+3x^2)dx#. (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^2-3x+6# in the range #2le x le8# (b) Use the trapezium rule with 10 strips to estimate the area bounded by the curve, #y=x^2-3x+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^2-3x+6#, the lines x=-2, x=8, and the x-axis. (d) By integration, determine the actual
9.
Evaluate #int _2^4 x^2 + 2x - 15\ dx#
10.
Find the value of k if #int_0^2(kx^3 – 3x^2)dx=16#
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